Was the Space Shuttle landing sequence executed manually or automatically?
Since that's really three questions, I'll address them separately.
1. Did the shuttle have or use autoland?
The space shuttle did have autoland capability, in theory. When the shuttle was being developed, it was imagined that flights could last so long that human pilots might get rusty, or have their ability to function under gravity impaired due to muscle atrophy from extended time in micro gravity.
Unfortunately, testing autoland on the space shuttle was a bit risky. If anything went wrong, the shuttle has no ability to disengage and go-around, and it might be too late for a human pilot to intervene and still make a safe landing.
They did try evaluating an incomplete implementation of the autoland on STS-3, where the lineup, final approach (outer glide slope), and preflare maneuvers would be performed by the autopilot, at which time the commander would take control and complete the landing.
The result was one of the worst landings in the history of the program. The autopilot was unable to maintain a stabilized airspeed, at first closing the speedbrake completely, then opening it again until airspeed was low, then closing again at 4000 feet which caused the orbiter to be too fast by the time commander Jack Lousma took over. He didn't really have enough time to get a feel for the aircraft or make corrections, so it hit hard and fast.
Afterwards, it was decided to discontinue testing of the autoland system, although the implementation was eventually completed.

Development of an automatic approach and landing system was terminated indefinitely due to the inability to implement an FAA [Federal Aviation Administration]-like powered aircraft certification program for an unpowered glider like the Space Shuttle. That is, an automatic system cannot be reasonably certified for an aircraft without a go-around capability in the event of an autoland system failure, because the only recourse in the Shuttle is to land rather than to execute a wave-off maneuver as in a powered aircraft. The practice of having to salvage a good landing out of a poor approach is not professionally accepted airmanship.
Further, the STS-3 experience was classified as a “late-takeover” action in which a failure of the autoland system in even closer proximity to touchdown could result in an upset too late in the landing phase to be reasonably recoverable. &quot;Monitoring&quot; the approach versus actually &quot;controlling&quot; it induces a time lapse in the mental-to-physical control conversion process which increases in criticality for manual takeover as the aircraft approaches touchdown. An automatic failure on the IGS, with no recourse for a wave-off, would be unsafe compared to a totally manual- controlled approach.

The quote is from NASA Johnson Space Center Oral History Project where Jack Lousma gives a great description of the flight and landing a little less than half way down the page.
The Columbia Accident Investigation Board Report did recommend testing the autoland system, but to my knowledge, it never was.
Just to clarify, the autopilot was used for most of reentry on every flight, but I believe your question was specifically about the final stages of flight and landing.
2. What sort of simulator was used for training?
So the shuttle was hand-flown for landing (a fly-by-wire system, so it's arguable that there's no true manual mode), and that means pilots have to be trained.
They were actually trained in several different simulators. There was SAIL (Shuttle Avionics Integration Lab) where they did serious integrated systems and software testing.
There were also fixed and motion-based simulators in Houston. These were collectively referred to as the SMS (Shuttle Mission Simulator). These were used for training for all phases of shuttle flights, not just landing. They were particularly useful for simulating contingencies, like an RTLS abort (return to launch site). It's similar to how simulators are used in the aviation industry to train for scenarios which would be too dangerous to train for in a real aircraft.

The motion-base SMS. Image Source
For nominal conditions, the best simulator for landing the space shuttle was actually another aircraft referred to as the STA (shuttle training aircraft). This was a modified Gulfstream II which would fly with its main landing gear lowered and thrust-reversers engaged to simulate the high drag profile of the orbiter. The left side of the STA cockpit was built to mimic the layout of the shuttle, including the heads up display.


Images from the STA's Wikipedia page
The STA was used for both training shuttle pilots and for evaluating weather conditions at landing sites on launch and landing days.
Lastly, there was also an on-orbit simulator that the pilots would use for practicing in the day or two before landing. It was basically a laptop with special flight-simulator software.
3. Is the simulator available in any flight simulators?
The answer depends on exactly what it is that you're asking. The actual software used on shuttle computers (written in HAL/S) is ITAR restricted, so it is not available to the public. Sadly, even the on-orbit simulator has never been released, to my knowledge.
However, there are several simulators which let you fly some or all of reentry and landing to varying degrees of realism.

SSMS 2007 is a pretty high-fidelity simulator, but has a horrible, not even remotely accurate, reentry and landing.
On phones and tablets, I love fsim Space Shuttle. It's decently realistic, but only lets you fly the subsonic portion of flight.
Orbiter, which isn't a shuttle-specific simulator, but there are shuttle plugins. I've flown several manual reentries in Orbiter, but you really have to know what you're doing or you won't make the runway. That's probably the best option if you're really interested in orbital mechanics, and a terrible option if you're not.
X-Plane has a Space Shuttle scenario, but it's not particularly realistic.
I know some people fly Shuttle missions on Kerbal Space Program, but I've never played with it at all so I can speak to it one way or the other.

Interestingly, the motion-based simulator from the SMS was donated to the Aerospace engineering department of Texas A&amp;M, and it was discussed making the simulator at least partially available to the public. Unfortunately, I think they ran out of money to actually put the thing together and run it, so it's effectively just sitting in boxes in a room somewhere. They have a facebook page, but it's been a long time since I've seen any progress.