What would be a good ratio spending time between slow games, study and fast games? I'd play like 90% of my games at slower time controls if speaking about a player under 1200 rating. The last 10% can be occasional speed chess if they want to have fun or experiment with shorter time controls to gain experience with it, but even then, the speed chess probably won't facilitate learning much.
Do fast games make you think quicker? (Seems like the blunder rate can be high) Yes and yes. This is the primary danger of speed chess. You don't want to condition your mind to move quicker because they you'll find it tougher to patiently think during longer time controls and you'll be prone to blunders. Even if you play longer time controls regularly, I would recommend playing one you are comfortable with and then occasionally playing a different time control just to help prevent conditioning you to moving at a single pace.
What do fast games teach you about time control? Similar to above, mixing up time controls can teach you about time management skills and help prevent you from moving at one pace only. However, fast games in particular don't teach you much about time control; this comes from the varying of time controls so keep aware of the clock remaining and how you manage it (especially under pressure while remaining calm).
Looking forward to hear educated opinions from more experienced and strong players. Happy to help. I'm sure most players will have similar views to me as many have said this about time controls too. What I am saying isn't new. It is just that the best speed chess players in the world are also the best chess players in the world at longer time controls. Learning chess at longer time controls translates some ability into speed chess, but lesser so the inverse as improving at speed chess doesn't always mean you have strong chess ability at longer time controls; they might just be quick at moving for that time control.
(My responses in bold above)
From what rating does playing blitz make sense to improve your game and why?
What would be a good ratio spending time between slow games, study and fast games?
Do fast games make you think quicker? (Seems like the blunder rate can be high)
What do fast games teach you about time control?
Looking forward to hear educated opinions from more experienced and strong players.