Theoretically with good spark and proper timing, the engine should at least fire up on starting fluid. Are you positive you have the engine timed properly? (It is possible to be 180 degrees out) It sounds like you do but worth a double check...
Those appear to be a set of Burgard aftermarket a arms. Aftermarket a arms were never made for the first gen warriors, and if they don't fit on the second gen warrior, then they are not warrior a arms. (Just by looking at them, I would say they aren't warrior a arms anyway...probably aren't even...
Couple things could be wrong:
a. you aligned the flywheel to the T on the flywheel, but not on the compression stroke, so the timing is off. When setting timing, you need to make sure the motor is at TDC on the compression stroke, since the there is a 2:1 ratio between the cam and the crank...
Most shock builders will be willing to cut down the 660 shock for you. The respring/revalve is standard procedure, so everywhere will do that. I don't have any personal experience having it done, so not going to recommend anyone, but if you search other ATV forums, there are tons of threads...
The 660 rear shock is too long to fit the stock linkage, and will slightly bind the suspension, and increase the chance of breaking your chain/chain rollers. The only shock I would recommend would be the banshee. Or you can get the 660 cut down. You're going to want to re-valve/re-spring...
It sounds like you could adjust the clutch a little to just take some of the slack out on the lever side. It's normal for these bikes to drive in reverse even with the clutch pulled all the way in, especially when cold.
Going to be tough to say whats wrong just based off of your post. Maybe a video will help? Either way, at 70-75psi, it needs a rebuild anyway, so i'd just start tearing into it
This stumbling off idle is a common problem on the warriors when you add an aftermarket air filter/exhaust without upping the pilot jet. They are slightly lean on the pilot stock so the added airflow causes the stumbling.
You take a tire valve stem cap and place it on top of the accelerator pump arm in the carb. This makes the arm that presses on the accelerator pump push it sooner and farther down, squirting more gas into the carb when you WOT it from a stand still. Apparently it helps, but I don't have it done...