I'm a Garen main, so I can't really comment on Fizz too much, but as Garen I have four potential roles; split pushing, looking for picks, backline diver in teamfights, or backline peeler in teamfights.
The only time I should peel is against combo based assassins/fighters, because outside of my execute I really only have a silence to help, so it's not a common role for me to take. If I'm forced to teamfight my main thing will be trying to find the largest threat to my team and burst them down, be it a tank, dps, or whatever. If my team can handle themselves my work is done.
My preferred role is split pushing, which goes hand in hand when looking for picks since I'm looking to always be moving around pressuring the side lanes. If I get a chance to flank at in a teamfight I'll gladly do it, but mostly I'll be taking towers and forcing at least two people to come stop me, which gives my team a better chance at teamfighting. Split pushing is all about being where you need to be at the right time, because applying pressure to bot lane when your team just backed to base doesn't actually help your team one bit.
The big thing here is to realize every player has a niche. Be it a pro player or one of us average scrubs we all have our own unique sets of skills, and we need to find champions that fit into those skills if we want to be effective. I prefer the macro play, so I play mechanically simple champions that allow me to focus on other things. I wouldn't say your role changes based on rank, but it definitely does with your personal skill. If your map awareness isn't great you shouldn't split push and should stick to teamfighting.
Fizz is a high burst assassin, but I don't feel like he's great at fighting multiple opponents without help. If you're going to roam bot I'd say do it alongside your jungler and make the gank a complete stomp, but roaming top can be great for getting a few solo kills. You're not much of a split pusher, but you can flank pretty well, so moving around the jungle warding/killing wards and controlling the space a teamfight will play out in is a great thing to do. Keep in mind there is a level of threat that comes from you simply existing, so you don't always have to be getting kills to be a positive force for your team.
Fizz really excels at one thing, so he's not as flexible in his role as other champions are.
Does the role change based on my champ, overall team comp, or matchup?
Yes, yes, and potentially yes. Just as with players each champion does certain things well and others poorly. Some do everything well but are not amazing at any of it, and some really need to stick to their strengths. Learning the limits of what a champion can do is part of mastering them, and sometimes quick tips just can't teach the same lessons as practical experience.
You've got a good mindset here, so I'll just wish you luck.
