1. Playing on LIVE with an exploited console is not something you can do more than once. Any exploited console going on LIVE will get banned within hours.
2. There used to be downgrading back in the day before a certain update outdated the CPU timing attack, but no longer. eFuses
3. I'm not sure about getting banned indefinitely, but I have heard of people getting suspended for using controller mods. My understanding is that they're against the Xbox LIVE TOS.
4. If the gamertag is infringing, typically that is what gets banned. If the console is modded or breaks TOS, that gets banned. In extreme circumstances, both can get banned.
5. Lots of people have multiple consoles, one for homebrew and one for LIVE. Some people don't like Xbox LIVE, so they're indifferent to being able to play. Some people still like single player games.