Questions about re-purposing double gang electrical box with two circuits
You are using the conduit wiring method with THHN wires*.  In this wiring method, 


Grounds are green, green/yellow, bare, or listed metallic conduit such as EMT
Neutrals are white or gray
Hots are black, brown, red, orange, yellow, pink, blue or purple


That's not news to you.  This is: NO SUBSTITUTIONS.  In things like switch loops and 3-way circuits, you often see a white wire used as a "hot". They're supposed to be marked with tape but nobody does. This is only allowed in cable wiring methods where you have no choice of wire color.  Not here!  In conduit with THHN, you must pull the correct color, period.**  You can mark them with tape, but doing so can't turn a neutral to a hot.

Another thing about conduit is the new "neutral in a switch loop" rule does not apply. That's because it's easy to pull a neutral wire in the future.  Switch loops will have always-hot and switched-hot. 

One rule remains the same: Currents must be equal among all wires in a cable or conduit.  This could also be called the "Tree rule": circuits can branch but one branch can't loop back to rejoin another. 

A side-effect of this rule is neutrals are not promiscuous: A neutral can only be a neutral for the circuit it's part of.  For instance you cannot use the lower-left white wire to provide neutral for the upper-right black wire; they are on different circuits and this will result in unequal currents in both conduits, as well as overload the neutral. Neutrals don't have circuit breakers! Their only protection is honest wiring: If a neutral returned two circuits' worth of power, it would overload. 

Any promiscuous/stolen neutral issue will also trip a GFCI protecting the circuit. 

To answer your questions briefly, #1 if it's white, it's neutral for the accompanying circuit only.  #2 That's just phantom voltage from running parallel to the hot wire.  Don't worry about it.  #3 NO because that would involve stealing neutral and violating the "equal currents" rule.

So on to your problem. Let's divide and conquer. 

The top right "circuit"

Current can't return from a load without a neutral.  And currents must be equal in each conduit.  And there's no neutral in the upper right conduit. That means it can only be one thing: A switch loop.  You confirmed this in comments.  A switch goes there. 

I imagine the red and black went to the two conductor screws on one switch. This is the only possibility without violating the "equal currents" rule, so it must be so.  If you feel the switch is useless to you, you can create an "always off" switch by capping both wires separately -- or an "always-on" switch by tying both wires together with a wirenut.   That would free up one yoke space in this box, for say a receptacle. 

You cannot alter this circuit without a lot more analysis and work.  Next. 

The left "circuit"

Reds went to the switch, whites nutted together.  Straightforward stuff.

If you wanted to fork a receptacle off this, make 3 6" pigtails: 1 white and 2 red, from #12 THHN stranded wire. The receptacle gets 1 white and 1 red pigtail. The switch gets 1 red pigtail on 1 switch terminal, the other terminal goes to the red from the upper left conduit.  Now tie all the whites with a wirenut, and all the dangling reds with a wirenut.  Voila. 

(This bundling by color is lucky, it's not a sure thing and can differ in different circuits.) 



* Conduit in residential is unusual.  Chicago requires it, mixed-use developments probably require it, large multi-unit buildings, or your electrician is picky!  I've seen Romex Kaisers fill conduit with only black, white and red THHN, thinking they must follow the exact same color code as cable, but those fools would pull black+white for a switch loop (illegal obviously) and that didn't happen here. 

** Except in very large sizes, 4 AWG and larger, generally used as feeder or service.