When most games are connected you could easily get away with a 15mbps connection or less, and still have no issues. I have 25mbps and zero issues when gaming other than the downloads taking a while.
Faster speed does mean faster downloads though. Increasing speed all depends on what is available in your area. If you don't have fibre in your area, your options are likely limited for now.
Your connection speed won't have anything to do with you being disconnected. That is a symptom of a different problem.