Electrical – Size of electric wire needed to serve overhead projector and screen off 20 amp breaker

I'm not sure where you live but US code calls for a minimum of #8 if you're using copper or #6 if you're using aluminum. There would be no advantage in using a larger wire size in this case because at 30ft you will have negligible losses.

I don't know what your voltage is or how many phases you have but for an example of what your losses may be, I will assume a worst case scenario - 100V single phase. If you used the #6 aluminum wire, you would have a maximum of 1.72% voltage drop (loss). If you used the copper #8, you would have a maximum voltage drop of 1.66%. And just remember, when you are calculating losses (voltage drop), the higher the voltage, the better because higher voltages have less voltage drop.

Note: I use the Southwire Voltage Drop Calculator for most of my voltage drop calculations. http://www.southwire.com/support/voltage-drop-calculator.htm

I will answer your questions in turn, and then add a few notes at the end -- the code for pools is complex, partly because they are bespoke installations for the most part and partly because of how pool electrical safety evolved over time.

1. You actually cannot do this -- the wire run must be in conduit as per 680.25(A)(1). You can use Schedule 80 PVC or RTRC-XW for the entire run provided the stub-up is protected from physical damage (i.e. being clobbered with a mower or whatnot -- 4x4 corner posts around the panel should do the trick, as would mounting it in a drained pump room or pit using an outdoor enclosure, provided clear working space is maintained). Otherwise, you're stuck with RMC or IMC (NOT EMT) -- likely red brass to avoid corrosion.
2. That 125A number is overkill, requiring a 1/0 or 2/0 triplex in Al. I'd recommend instead using individual XHHW-2 Al conductors instead of a triplex (with an 8AWG copper THWN-2 or XHHW-2 wire for the EGC, of course) -- this will be easier to get through the conduit than a triplex, and also isn't liable to short the neutral to ground unexpectedly if you're running it in red brass. (Conductor/subpanel sizing advice is below, if you wish it.)
3. You are correct -- never bond neutral and ground in a subpanel.
4. Your overall plan seems sound, provided the subpanel is correctly placed (it cannot be within 5' of the nearest edge of the pool).

Additionally, given the type of pool you are dealing with, there are a few additional points to consider:

1. You'll need a convenience receptacle somewhere outside 6' but within 20' of the pool -- this can be at the subpanel, attached to the lighting circuit and protected by its GFCI. (While you don't need GFCI protection on a lighting circuit that's feeding listed low-voltage ground-free luminaires -- it's cheap insurance.)
2. Your branch circuits will need to be run in conduit as well -- either red brass RMC or IMC, or Schedule 80 PVC or RTRC-XW if you wish to use nonmetallic conduit. In any case, you must run an insulated EGC with the branch circuit.
3. EQUIPOTENTIAL BOND ALL THE THINGS. If it is conductive, not a hot or a neutral, and is within 5' of the pool, don't take a chance -- just hook it up to the equipotential grid. In your case, since your pool is vinyl-lined, you do not need to run the grid under the pool, just as part of the perimeter paving -- 12" O.C. 8AWG bare copper is used for at least 3' beyond the sides of the pool, with cadwelded (i.e. exothermically welded) or otherwise listed connections. You'll also need to hook this grid up to the pool pump motor's ground wire (which is where it'll tie into the electrical system's equipment grounding system -- for a double-insulated pool pump, just run an EGC down to the pump's j-box as normal and tie the equipotential bond wire to it there), and to an underwater equipotential bonding electrode made from corrosion-resistant materials that has a minimum of 9 in^2 of surface area and can't be damaged by pool activity if there is no bonded metal in contact with the pool water. Also, I recommend running another bare wire from the grid to the subpanel itself, directly -- this is insurance against the EGC connection at the pump motor getting undone.

As to that forum that told you not to bond the equipotential grid to the house grounding system, whoever was posting that there is full of baloney, to put it extremely mildly. NEC 680.26(B)(6)(a) explains it most clearly (strong emphasis mine):

(a) Double-Insulated Water Pump Motors. Where a double-insulated water pump motor is installed under the provisions of this rule, a solid 8 AWG copper conductor of sufficient length to make a bonding connection to a replacement motor shall be extended from the bonding grid to an accessible point in the vicinity of the pool pump motor. Where there is no connection between the swimming pool bonding grid and the equipment grounding system for the premises, this bonding conductor shall be connected to the equipment grounding conductor of the motor circuit.

Finally, as to that 125A number, it's hooey -- the pool pump motor is 8FLA@240V as per Table 430.248 and uses a 20A breaker from table 430.52, not the whopping 60A fib that you were told. In fact, that 60A number that they're giving you just for the pump is enough to leave 1200VA (600 per leg) for lighting and convenience receptacle loads after the 10A max load from the pool pump and the 40A number given for the heat pump -- which is well over three times the 180VA/outlet the Code tells you to allocate to general receptacles and outlets that aren't covered by other loads.

With that information in hand, we can run 4 6AWG THWNs for the feeder, just as you propose, and 14AWG for the pool pump as 430.22 specifies a conductor rating of 125% of FLA, or 10A in your case. (You can also use 14AWG for the receptacle/lighting circuit -- nothing says it has to be 20A.)