Wiring from garage panel to welder

NEC 630.11(A) may be of interest.

National Electric Code 2008

ARTICLE 630 Electric Welders

II. Arc Welders

630.11 Ampacity of Supply Conductors.

(A) Individual Welders. The ampacity of the supply conductors shall be not less than the I1eff value on the rating plate. Alternatively, if the I1eff is not given, the ampacity of the supply conductors shall not be less than the current value determined by multiplying the rated primary current in amperes given on the welder rating plate by the factor shown in Table 630.11(A) based on the duty cycle of the welder.

Also take a look at 630.12 Overcurrent Protection.

The Owners manual (PDF) seems to contain the answer.

Wiring from service panel to sub panel

If we look at 110.14(C) in 2008 NEC, it says:

National Electrical Code 2008

ARTICLE 110 Requirements for Electrical Installations

I. General

110.14 Electrical Connections.

(C) Temperature Limitations. The temperature rating associated with the ampacity of a conductor shall be selected and coordinated so as not to exceed the lowest temperature rating of any connected termination, conductor, or device. Conductors with temperature ratings higher than specified for terminations shall be permitted to be used for ampacity adjustment, correction, or both.

Which means we'll have to check the breakers temperature rating, and the subpanel terminals temperature rating. Then we'll have to look at table 310.16 (Table 310.15(B)(16) NEC 2011), so we can determine the wire size we should use.

In this example image you can see if we're using copper, we'll use 60°C. Once we look at the table, we'll see that for 50A @ 60°C we need 6 AWG copper wire. (50A @ 75°C = 8 AWG).

110.14(C)(1)(a) basically says that if the temperature is not listed, use 60°C for circuits less than 100A. 110.14(C)(1)(b) says if the temperature is not listed, use 75°C for circuits over 100A. However, in the real world 75°C is often used for both.

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.)