How can I increase white wine shelf life specifically bought for cooking?

Freeze it in cubes and use it as required. Wine should be low-alcohol enough to freeze in a regular freezer, although you might find you get 'slushier' parts – if you do, these will have a higher alcohol concentration – so be careful about the container you use.

To answer your other questions, wine going 'bad' is a taste issue rather than a food safety issue, so if the wine tastes fine to you it is fine to cook with and if it tastes bad to you I wouldn't cook with it. Sure, you might put sour wine into a recipe and notice the sourness less than if you just drank it, but it's certainly not improving your food at that point and I wouldn't use it.

I will say that when people say wine lasts only a few days they're often taking from the perspective of someone with a very particular palette, and if you're aiming at 'tastes like wine' you'll likely be fine for at least a week, keeping it sealed in the fridge to slow oxidation.

Small bottles.

These bottles are 187 ml which is 6 ounces each. This 4 pack costs $2. You will use one bottle to cook and discard what you don't use from that bottle. The other 3 will wait their turn and stay good while they do.

You will not find fancy wine in 6 oz bottles. You will find cheap wine. That is the kind of wine you should be cooking with.

There are 'bag in box' wine packages around, easy to find in some places but harder to find in others.
With those no air comes into the container keeping the wine good for a much longer time.

They were almost standard in Australia when/where I visited back in 2005, I am not sure how much they have spread around the world.

For the sake of posterity, I'm going to make one more suggestion here:

Buy moderately priced wine with a screw cap, use what you need, and leave the rest, tightly closed, in the pantry.

If your pantry is kept temperate (15-24C), the half-used bottle of wine will keep for months to a couple of years. It won't be good enough to drink, but it will be OK to cook with.

Occasionally wine turns to vinegar or goes strongly off in ways that prevent cooking with it. I find that it's sufficient to keep a backup bottle of cooking wine for that. Yes, freezing wine, refrigerating it, or vac-packing it will allow it to keep for longer, but frankly aren't necessary unless you live somewhere tropical.

I drink wine, and keep leftover half-bottles to cook with, and have been doing the above for years without real issues.

Alcohol that is exposed to oxygen in the air will turn to vinegar, this is what stops it being drinkable. A bottle at room temperature can be left open for 12-48 hours, depending on a person's taste for tart vinegary things. If you close a partially drunk bottle, it should be fine to drink for 2-7 days on the same vinegar taste preference basis. Refrigeration will push everything here significantly longer, and anything that either prevents new air getting to the wine (any kind of lid), or sucks out air (like a vacuum pump cap) will also significantly extend the life span.

Wine that has turned to vinegar (but kept closed) should actually keep for quite a long time. I often have a closed half bottle of wine knocking around the kitchen for over a month before using it. If the wine goes bad, it should be visibly cloudy or have an obvious off smell (more than vinegar). If you really need to be sure, adding salt to the wine should make it last longer, say at the rate of 1/2 -3 tspn to a full bottle (3 is pretty damned salty). The problem with adding salt to the wine is that a) you have to remember & adjust future recipes based on this, and b) any recipes that call for reducing cooking wine will be very very salty.

I happen to like tart or vinegary food, but I do not at all like tart wine. This means that I often have & happily use old wine in cooking and have maybe only once in my life thrown away wine.

Final note : the reason why cheap /low quality wine is used for cooking is that the "fine flavours" of top notch wine evaporate very easily. This is both why low alcohol wine is difficult to make very good, and why good vs bad wine for cooking makes little difference.

Wine goes off because it oxidises, you can avoid this by pumping the air out of the bottle using something like Vacu Vin (I presume cheaper alternatives are now available and probably work fine, that's just the brand I've used). These devices consists of a rubber "cork" with a valve that goes into the wine bottle, and a pump which allows you to pump (most) of the air out of the bottle. Thus pumped and sealed the wine will keep for weeks without noticeably degrading, especially when refrigerated.