1. Nope. Basically you rise and shine, wake up and smell the ashes, and promptly get thrown into the future. The entire experience is new to Gordon Freeman too, since he was in a 20 year stasis between Half-Life and Half-Life 2. (You'll be able to play the 'Black Mesa' fan remake of HL1 for free because you'll have Source engine games.)
2. HL2, Episode 1, Episode 2. Hurr.

3. YES. A MILLION TIMES YES.

Every class update (of which there have been several) has been 100% free. Valve gives you the Source SDK when you buy The Orange Box (or basically any official Source engine game), so there are hundreds upon hundreds of free custom maps of varying quality. (Some like Badwater Basin are so good that they were officially adopted; some like any of the Orange maps are so bad that they're mocked by everyone but newbies and 12 year old Counter-Strike kids.)
4. No. TF2 does have bots, but it has no AI so they're really only useful for machinima and target practice. You can't play realistic games against them.
5 YES!!!11!2
PC TF2 >>>>>>>>>>>>>>>>> Console TF2. With payload, arena, payload race, new weapons, and custom maps, it's like a completely different game. The 360 might get a pay-for update pack sometime in the future (they said just the launch TF2 was pushing the 360's memory limits), and to my knowledge there are absolutely no plans of releasing the DLC on the PS3 because Valve doesn't have internal PS3 developers and EA (who ported it in the first place) won't work for free. You also add to the value when you get the PC version because you get access to dozens and dozens of free Source engine mods. (Garry's Mod is great too and it's only $10 but you need a Source game to play it.)
In other words:
PC Orange Box =
Console Orange Box =
