My display feels weird when I utilize a lower resolution than its natural output. Is it better to buy a 1080p monitor to game at 1080p or could you do it at a 4k monitor without any quality issues?
I'd ask you to get a 1080p monitor if all you're going to do is 1080p gaming. Yes you can game on a 4K display at 1080p but the pixelation in itself will want you to either go for a smaller display(1080p res) or up your hardware specs so you can game at 4K). The latter option is expensive, which is why I'm suggesting to stick to 1080p displays.
I can't. They don't make 1080p OLED display. I need OLED because only TN Panels with their awful colors offer high response time/refresh rate. BUT, OLED panels offer best of both worlds. They used to come in 1080p models but no more.