Code P0302, engine miss?

It will clean the injectors, you shouldn't have too much of an issue with carbon on the intake valves with port injection. Since the injectors are spraying on the back side of the intake valves, that's pretty much how they stay clean. The newer vehicles with direct injection (GDI) are the ones that have serious carbon build up issues. So just keeping up with fuel system cleaner and fuel filters you should be okay.

It will take more than a gallon if you're draining the system, you can buy premixed coolant so you don't have to worry about what might be in the local water. If it's running warm going up mountains, the thermostat is probably gummed up since the coolant hasn't been serviced in so long. Thermostats can get to a point where they will stick at a certain range of movement, but if you flush the system with water, you're going to end up with just plain water in the engine block, it won't all come out, So just pull the lower radiator hose off, let it drain and refill it. That way you don't end up with a diluted mix. Sounds like you live in a wintery place. The spec on coolant capacity is 13qts for the entire system (3.25 gallons) so I'd get 3 to be sure you have enough.

Well, running anything through a higher pressure will dislodge whatever is stuck in there. The cooling system only runs at around 18 to 20 PSI max, Thats why there is a max pressure written on the radiator cap for pressure testing. Mostly what builds up is an acid type of goo which is a reaction from the electrical system. Here is basic guide on draining and flushing, then refilling the system. There is a video too on filling it with plain water and running it, that will hopefully help clear out what's stuck in the thermostat.

https://www.2carpros.com/articles/coolant-flush-and-refill-all-cars

Honestly, I have never used seafoam, I heard a few bad things about when it came out, so I just avoided using it at all. If you looked under the valve cover and there was no carbon build up there, I'd say you're good to go. That is the place that will have carbon build up the worst in most cases because of the PCV system. Just keep up with the oil changes, and the synthetic is a good idea as well. Sounds like you don't drive the vehicle that much, every 2k miles is even a bit early for oil changes, but it certainly doesn't hurt anything. You can check the PCV valve and make sure that it will only allow air to flow in one direction. They tend to get carbon build up and stick open, which leads to carbon build up in the intake manifold and the throttle plates, and every other place you can think of. So, swapping that out when doing a tune up is good.
But it sounds like you're on top of maintenance very well.

How long has it been since you've had any A/C work done? Vehicles this old will eventually develop a leak somewhere. Usually if it's something that has not been recharged in 20 years for example, and the system has slowly lost its charge and it's not a seal that completely failed, it's usually the fill valves that leak slowly over the years. You can add a can, or add half, let the system stabilize, see how cold it gets and go from there. Keep the A/C controls on the recycle or defrost setting so it doesn't draw in outside air. If you own a set of gauges it doesn't hurt to see what the pressures are at, with those sets the yellow center hose is to add refrigerant. That being a 1996, make sure to check the label first, in service info it looks like its R134A, but just look for a label. This is the pressure to temperature chart below for R134a if you do hook up the gauges.
And depending where your fill valve is, it's going to be on the Low-pressure side, sometimes companies put the fill valve on the compressor itself, yours shouldn't be, here is a guide to help recharge from those cans. If you're filling with a valve that is on the compressor, you just don't want to put liquid freon into the compressor. Turning the can upside down adds the Freon as a liquid, and right side up adds it as a gas. Just so you know about that. If you have one of the cans with a gauge on it, and the pressure is zero, then the system is completely empty, but it's probably like my truck, where it gets semi cool on really hot days. It's just low. Add some and then check how cold it gets, then if needed add a little more.

https://www.2carpros.com/articles/air-conditioner-how-to-add-freon

Well, if you want it to last another 10 years, the best thing to do is have a shop evacuate the system with an A/C machine. This way they can measure how much is taken out. If it's low on the amount that should be in there, then there is a leak somewhere and it should be found and repaired. That way the new refill will last, and you won't have to refill it every year. Because there's no way of telling when the leak actually started, so if it's a fill valve for example that section of line with the valve can be replaced. The fill valves are almost exactly the same as a valve stem on a tire. The one-way valve has those small seals on them that leak. If you have the vehicle off, and you put your ear real close to the valve while taking the cap off, you can almost always hear a small hiss of a leak. These leaks are overlooked most of the time because when the A/C machine is on both fill valves (High side and Low side), the leak is not detected. You can also spray each fitting of the A/C lines with soapy water in a spray bottle, and the leak will bubble up. Dish soap works the best.
If the system is still running semi cold and the compressor is still cycling, you can find a leak with soapy water.
Most leaking areas are any connections, the front AC condenser in front with the radiator (they leak a lot at the soldered bent connections), fill valves, compressor seals.
You can find the leak, do the repair and then have a shop vacuum it down and refill it. Or you can just add a can and see how long it lasts. So there's a couple options here. I'll bet you can find a leak pretty quick. If you do, take a video of it for us. Here's a soapy A/C leak picture. Sorry for the long post.

No don't add any oil, the only time you add oil is when replacing a component, you drain the oil out of the part being replaced and add that exact amount back in. For a refrigerant leak, you just add the correct type of Freon. This guide will tell you everything you need to know, just add a little bit at a time while monitoring how cold it gets.

https://www.2carpros.com/articles/air-conditioner-how-to-add-freon

The specs for your vehicle is quite a lot of refrigerant. 2.0lbs or 2.25lbs. With 16 ounces in 1 pound, that's 32 ounces to fill the system for a pickup model. So, you can buy one of the 13 ounce cans and just add it slowly, the guide has you raise the rpm while filling it a little at a time, until it gets cold enough for you. I bought a 13oz can from harbor freight for $12 the other day. So, if you still have semi cool air, you shouldn't need to add more than 1 can, just go slow while adding it, give the system time to adjust and build pressure before adding more.