That depends on how far back in the day you want to go.  Amphetamines were massive in baseball all the way back to at least the 60s and, according to some, before that, while steroid use began to grow pretty fast during the 70s.  Even going way, way back to the 20s, there are accounts of players attempting to use testosterone serums, though they probably weren't very effective.

The only real difference between then and now is that the stuff now works a hell of a lot better.  Some of the most honored and respected players in baseball history have admitted to using amphetamines, for example.

As for A-Rod, I think this fits with what other players have gotten.  He shouldn't get more just because he's more famous.  I'd like to see him out of baseball but they can't just single him out.

As for taking stats and records away, at this point that's pretty much an impossible task.  If they started that route they'd have to go all the way.  Baseball has a very long history of corruption, cheating, and drug use.  If they really started digging, half the HoF would be wiped clean.

It varies greatly.  Some, especially Latin American players, will start in the minor leagues because they feel they need an edge to get them to the major leagues.  A lot of players have said they started when another player introduced them to a source and started the "everyone's doing it" routine.

I think the most common situation, though, is when a player gets injured or worn down and they start worrying about their future.  They take it so they can stay strong for the entire season and bounce back faster.  That's why a lot of pitchers take it, especially.