How do pilots use stopway, EMAS, clearway and displaced threshold?
This is a loaded question.  Every regulation in aviation seems to be written blood and the need for runway safety areas (RSA) is no exception.
RSAs are defined in AC 150/5300-13A

Runway Safety Area (RSA). A defined surface surrounding the runway prepared or suitable for reducing the risk of damage to aircraft in the event of an undershoot, overshoot, or excursion from the runway.

RSA's were developed for specific reasons and are discussed in section 307 of the AC.

In the early years of aviation, all aircraft operated from relatively unimproved airfields. As aviation developed, the alignment of takeoff and landing paths centered on a well-defined area known as a landing strip. Thereafter, the requirements of more advanced aircraft necessitated improving or paving the center portion of the landing strip. While the term “landing strip” was retained to describe the graded area surrounding and upon which the runway or improved surface was constructed, the role of the landing strip changed to that of a safety area surrounding the runway. This area had to be capable under normal (dry) conditions of supporting aircraft without causing structural damage to the aircraft or injury to their occupants. Later, the designation of the area was changed to “runway safety area” to reflect its functional role. The RSA enhances the safety of aircraft which undershoot, overrun, or veer off the runway, and it provides greater accessibility for fire-fighting and rescue equipment during such incidents.

Runway safety areas surrounding a runway is typically twice the width of the runway and 1000' longer than the runway on both ends.  The 1000' of additional length is referred to as a runway end safety area (RESA).  Objects inside the RSA must have specific function and be frangible (fall down) if hit by an aircraft)

AC 150/5300-13A Figure 3-5
Let's talk about runway end safety areas (RESA).  The 1000' feet of additional space beyond the end of each runway is now mandatory.  Every airport must have a RESA at the end of the runway.  The only exception to that rule is runways with an EMAS installed.  The 1000' is not an arbitrary number either.  Through analysis of aircraft overrun incidents from 1975 to 1987 it was determined that approximately 90% of the overruns stopped within a 1000' past the end of the runway at speeds less than 70 knots.  See AC 150/5220-22a for reference.
There are airports that cannot comply with the RESA requirement due to topography, roads, railroads, houses, etc.  In these situations, the FAA has established a concept called declared distances that will push the RESA onto the runway.  This will limit the accelerate-stop distance available (ASDA) and landing distance available (LDA) values.  Airports that cannot comply with the RESA standard are limited to a runway length which is available and suitable for preflight planning purposes.  It would be impractical for the FAA to tear up perfectly good runway.
This means pilots must reference declared distance information from the current chart supplements booklet to take advantage of runway safety protections.
Take a look at KSAN runway 09 end.

Google imagery from KSAN Airport
A 1000' RESA lands in the middle of shopping center.  In this case, the RESA is moved onto the runway.  The pilots must reference the chart supplements to determine usable lengths for takeoff and landing.  Here is the declared distances from the chart supplements booklet.  The runway length is 9400' but the ASDA and LDA values 1,121 feet shorter due to no space for a RESA at the end of the runway.

Chart supplements information for KSAN
Other parts of a RSA include a runway protection zone (RPZ).  This area extends up to 1700 feet past the end of the runway and can be as wide as 500 feet depending on the type of aircraft the runway was designed for.  A little history on RPZs

Approach protection zones were originally established to define land areas underneath aircraft approach paths in which control by the airport operator was highly desirable to prevent the creation of air navigation hazards. Subsequently, a 1952 report by the President’s Airport Commission (chaired by James Doolittle), entitled The Airport and Its Neighbors, recommended the establishment of clear areas beyond runway ends. Provision of these clear areas was not only to preclude obstructions potentially hazardous to aircraft, but also to control building construction as a protection from nuisance and hazard to people on the ground. The Department of Commerce concurred with the recommendation on the basis that this area was “primarily for the purpose of safety and convenience to people on the ground.” The FAA adopted “Clear Zones” with dimensional standards to implement the Doolittle Commission’s recommendation. Guidelines were developed recommending that clear zones be kept free of structures and any development that would create a place of public assembly.


AC 150/5300-13A Figure 3-17
Take a look at KSAN runway 9 again and you will see that the runway does not have space for a runway protection zone at the end of the runway.  A RPZ is mandatory and so the RPZ is moved onto the runway.  The takeoff runway available (TORA) is also 1,121 feet shorter than the runway for just that reason.
For takeoff performance calculations, pilots are required to ensure the airplane takes off and climbs to at least 35 feet within the TODA distance to achieve the designed safety in the RPZ.

Clearways are a special type of RPZ that will not allow any obstacles other than lights that are less than 26 inches in height.  The clearway must be at &quot;&quot;least 500 feet wide centered on the runway centerline and the length may not be more than 1/2 the runway length.&quot;

A clearway increases the allowable aircraft operating takeoff weight without increasing runway length.

In other words, takeoff distance available (TODA) is increased beyond the runway when a clearway is installed.

Displaced thresholds are created for four reasons that are beyond the power of the airport owner / operator.

A means for obtaining additional RSA prior to the threshold.  See
paragraph 307.
A means for obtaining additional runway object free area (ROFA) prior to the threshold.  See paragraph 309.
A means for locating the RPZ to mitigate unacceptable incompatible land uses.  See paragraph 310.
Mitigation of environmental impacts, including noise impacts.


Let me answer your questions to the best of my ability
What is the difference from a regulation point of view?
A stopway is designed to allow an aircraft to stop during an aborted takeoff.  Interestingly enough the AC 150/5300-13A does not say it is designed for runway overruns during landing or undershoots on the opposite runway.  By design it will increase the ASDA value for the runway
An EMAS system replaces a stopway and is usable for undershoots while landing on the opposite runway and overruns during both takeoff and landing and is not intended to meet the definition of a stopway.  An EMAS will not increase the ASDA value for the runway

How does the pilot know if this is a stopway where braking is desired, or an EMAS where thrust reversers might be hazardous.
The chart supplements book contains information on all EMAS systems.  Looking at the declared distances for the airport will indicate the presence of a stopway when the ASDA value is greater than the runway length.  Take a look again at KSAN and you will see a note for an EMAS off of runway 27.

Chart supplements information for KSAN
Stopways are a little harder to notice but can be spotted by noticing the ASDA value is greater than the runway length.

How do pilots use non regular runway portions such as clearway, stopway, EMAS, displaced threshold, and possibly more of this kind if it exists and is there some regulation which limits them for taking off, taxiing, or landing?
What are Runway Declared Distances?
This is the easiest question of the bunch.  Pilots are required to use the applicable declared distances for takeoff and landing performance calculations.  By design declared distances take into account all of the airport design standards such as clearway, stopway, EMAS and displaced thresholds.
Declared distances bridges the gap between runway design standards and aircraft performance.

§91.103   Preflight action.
Each pilot in command shall, before beginning a flight, become familiar with all available information concerning that flight. This information must include—
(a) For a flight under IFR or a flight not in the vicinity of an airport, weather reports and forecasts, fuel requirements, alternatives available if the planned flight cannot be completed, and any known traffic delays of which the pilot in command has been advised by ATC;
(b) For any flight, runway lengths at airports of intended use, and the following takeoff and landing distance information:
(1) For civil aircraft for which an approved Airplane or Rotorcraft Flight Manual containing takeoff and landing distance data is required, the takeoff and landing distance data contained therein; and
(2) For civil aircraft other than those specified in paragraph (b)(1) of this section, other reliable information appropriate to the aircraft, relating to aircraft performance under expected values of airport elevation and runway slope, aircraft gross weight, and wind and temperature.

The AIM has information concerning runway lengths.

4−3−6. Use of Runways/Declared Distances
4-3-6(c)(1) Declared distances for a runway represent the maximum distances available and suitable for meeting takeoff and landing distance performance requirements. These distances are determined in accordance with FAA runway design standards by adding to the physical length of paved runway any clearway or stopway and subtracting from that sum any lengths necessary to obtain the standard runway safety areas, runway object free areas, or runway protection zones. As a result of these additions and subtractions, the declared distances for a runway may be more or less than the physical length of the runway as depicted on aeronautical charts and related publications, or available in electronic navigation databases provided by either the U.S. Government or commercial companies.
4-3-6(c)(2)(c) When considering the amount of runway available for use in takeoff or landing performance calculations, the declared distances published for a runway must always be used in lieu of the runway’s physical length.
4-3-6(c)(2)(d) An aircraft is not prohibited from operating beyond a declared distance limit during the takeoff, landing, or taxi operation provided the runway surface is appropriately marked as usable runway.


AIM Figure 4-3-6 
How do you use declared distances for multi-engine pilots?
How do runway declared distances affect my takeoff distance?
The short version is aircraft utilizing balance field length calculations typically provide a single distance for takeoff that must be lower than both the TORA and ASDA value.
Aircraft with unbalanced field length calculations typically provide separate performance calculations for accelerate-go and accelerate-stop. The TODA and ASDA values must be greater than the calculated accelerate-go and accelerate-stop distances, respectfully.

TL;DR version
Pilots using applicable declared distances are afforded designed runway safety area protection.  Pilots who choose not to adhere to runway declared distances risk getting outside the protections designed for the runway.