are there are any e-mail services that are 'bad' to use in this case? (i.e. hotmail, gmail, etc.)
This isn't a practical concern. If those companies are breached in such a way that they gain access to your online gaming account, you've probably got bigger problems. As long as it's a reputable email provider, you don't have much to worry about.
should I avoid signing up to different services with the same e-mail address if I want to keep my accounts safe?
It certainly helps. If you use a unique email account for your online gaming account then not only will that make the username more secure, but it'll also make it much more obvious if someone tries a particularly good scam on you, since no matter how accurate it may be, it won't be sent to the address that official correspondence will be going to.
How to avoid being hacked as best as possible:
It's impossible to eliminate the risk entirely, no matter how knowledgeable you are. But the fairly basic steps above will massively reduce the chances of it happening.