Topic: PS4 games playable on PS5 ?

Hello, I own a PS4 Slim and today I bought a Samsung 55Q80T 4k TV. I understand that PS4 Slim does not support 4k, so I want to change it with a PS4 Pro or wait a little longer and buy my PS5 in November. I have 3 digital games bought and installed (Pes2020, GTA5, WOT) the rest of the games, around 100 games, are only bought, but not installed. My questions are:

1. If instead of PS4 Slim I buy PS4 Pro, and on PS4 Pro I log in with the same PS account:

a) do I find my games, or do I have to buy them again?
b) Do I find my progress online in each of those 3 games? (For example at Pes 2020 I have the team I built in the last year, at WOT I have several tanks that I bought in about 4 years, about the same at GTA)

2. If in 5 months I buy my PS5:

a) can I play the 3 games I bought on PS4 Slim? And if so, will they be at 4k resolution?
b) if question a) is affirmative, will the progress of the 3 games from PS4 to PS5 be kept?

@shaft29 answers to your questions where I can.
1a) No you don't have to buy them again. If you own them digitally you can just download them after you log in with the same PSN account on PS4 Pro
1b) As long as you have PS+ your saves should be in the cloud (double check). If you don't have PS+ you can also use an external hard drive to backup the data and transfer it that way. (It may be worth doing this anyway to be safe AND to backup other things like screenshots (if you use them) and settings

Note: they haven't announced full details of PS5->PS4 backwards compatibility yet but I'll do my best.

2a pt 1) Yes you probably can play the games on PS5. Though we don't know 100% for sure for all games but the likelihood is yes. (There may be a few PS4 games that don't run on PS5, or they may have to patch the game. They haven't announced this yet so can't be certain. But we expect most games to run)
2a pt 2) Will they be in 4K? It depends on the game. If the PS4 Pro patch exists and is 4k then yes. But many PS4 Pro games aren't 4k. They will likely be at least as good as PS4 Pro but they will likely not upscale to 4K automatically (as they haven't announced this and it would be a killer feature). Some games may be able to use the additional power of the PS5 to do things like faster load times but we just don't know for sure yet . Some games MAY offer a new PS5 patch to run even better but I wouldn't expect it.
2b) Again we don't know 100% for sure but almost certainly YES the game saves will transfer. Almost certainly via the cloud if you have PS+ and maybe via backup and a hard drive like in PS4 slim to Pro as above.

One last thing. Personally I really wouldn't recommend getting a PS4 Pro right now. I have both and Pro is good but it is relatively minor upgrade for most games and we are so close to next gen, I would definitely recommend waiting 3-4 months for PS5 if I were you. That's the smart decision.

Hope that helped