For the basics, first there was D&D (or Original D&D, OD&D). Following that was Advanced D&D (AD&D), with the two running together at the same time. AD&D had a second edition, which is generally referred to as just "2nd edition". I don't believe OD&D had a second edition at any point in time.
1.) "2nd edition", as mentioned, is referring to the 2nd edition of AD&D. AD&D and AD&D 2nd ed are pretty similar and somewhat compatable, in the sense that you could
use material from one in the other (with some work). D&D 3.5/Pathfinder might be a good comparison of the similarities and differences between the two systems. The three core books for 2nd ed. are the Player's Handbook, the Dungeon Master Guide, and the Monster Manual.
2.) OD&D and AD&D 1st edition are two different systems. When people are talking about "1st edition", they probably mean AD&D (in comparison to 2nd edition AD&D). The three core books for 1st ed. AD&D are the Player's Handbook, the Dungeon Master Guide, and the Monster Manual.
I actually haven't played OD&D and haven't seen the books, so don't really their names. A google/wikipedia search should clear things up, though.
3.) OD&D would have to be the system where races were classes. I just looked through my AD&D 1st ed PHB, and there were full rules for choosing a race and a class (complete with multiclassing).
4.) AD&D 1st and AD&D 2nd were not dramatically different - as I mentioned earlier, you could use material for one functionally in the other. The biggest change was all the different and various rules added to AD&D 2nd.
AD&D 2nd did also add several rulebooks towards the end of its life, called Player's Options, that added a lot of optional rules and abilities. I've never read through Player's Options, but there are people who consider the changes enough to call it a change in the game - a "AD&D 2.5e" so to speak.
5.) I'd recommend playing either AD&D or a free retroclone. Characters are easier to make and easier to die off, and there tends to be more focus on characters with humble beginnings than 3.5e's "everycharacter is special" focus. (Meaning: first level characters SUCK
) Monsters aren't setup to be level-appropriate, and you are expected to outthink and outfight opponents rather than making them "good challanges" for your level.