I'd put the chlorine tabs in -- A FLOATER, not directly -- just to use them up. They don't store well.

Otherwise, don't bother, till you get the test kit and test the water. The chlorine demand is going to be much greater than the amount of chlorine you have, so ultimately, it won't have any effect on the pool. But,you do NOT want to keep storing that tri-chlor.

Open those containers OUTSIDE, with you standing up wind, and holding your breath. If they've been sitting for over a year, whack the buckets HARD with a broom handle at arms length BEFORE you move them. Sometimes they can build up enough nitrogen trichloride -- which is a shock-sensitive but weak explosive, among other things -- when they are stored for a long time. If it's present, you want to 'blow the lid off', literally!, before you pick up the buckets.

Don't bother with the polyquat. It will be useful later, but it doesn't have the effect on surface tension cheap algaecide does, and that's what will kill your mosquito larvae.

Ditto on buying bleach, and stuff. A couple of gallons will only waste money.

Austin, TX climate info (from Wikipedia) suggests you don't need to worry about having winterized pipes and valves till December, so focus on the cover and the larvae. But, if the pump is not running . . . don't start it. By the way, you might want to consider operating budget effects from the pump: 2KW per hour, x Austin's electical rates, is a non-trivial expense and is far more than your 'operating' budget.

So, other than getting a cheap tab floater so you can use the tabs, just follow directions above, and worry about winterized pipes later.

But, if the pump hasn't be run in a year or more . . . you may not need to worry about that, either. Equipment damage has probably already been done, and will cost hundreds if not thousands to repair. Bad for the pool, but good for your CYA (Cover Your . . .) needs.