Though you don't have to, right? You can just jump right into the end scenario if you want (which admittedly isn't everything since there's new stuff in the main game as well) as far as I know.
Yeah, you can just play "The Answer" (the second scenario) without replaying through the first part ("The Journey"). Like you said though, there's new stuff in the main game and a lot of people want to see that. The second scenario looks cool, but it's aparently a pure dungeon crawl (in a completely different dungeon); no S-links or anything, which is both good and bad by the end of the game. I don't know if you can play it without a completed game save from the first scenario though.
So should I only battle on the freebie nights? Or should I battle every night, wait till my party gets tired and leave as soon as they do?
Well, I wouldn't go all out either way. I mixed it up a lot, going to Tartarus when I felt like it (and generally when I was in great condition), and buidling up my academics, charisma, and courage on the off nights. A lot of people with the original would only go to Tartarus on the last night, but I don't really suggest that either. I always figured if I did that there would be a good chance I'd get stuck without healing items or something like that.
In the end, it doesn't make a huge difference. The thing is that you want to have your academics, courage, and charisma hitting certain points by certain dates (more or less) in order to open up new social links and get good marks on exams (which nets you items I didn't use until the very very end of the game and probably could have gotten away without using them at all). You also generally don't want to waste a lot of time during the day building up these stats, as that takes away from valuable social link time.
It still isnt a huge deal unless you want to try to max out as many S-links as you can (I didn't even come close) but for the last like 2-3 months of gametime I literally had nothing to do at night. There was only one nighttime S-link in the version I have (I think there are two in FES), and after your stats are maxed and that s-link is maxed it's basically go to Tartarus or sleep.
It all evens out pretty well, since as you level up you tend to get tired less. I was generally as far as I could go in Tartarus each month after 1-2 nights spent there. Everything beyond that was basically just leveling up and building persona for fusing. The difficulty level in Tartarus scales kind of weird in that at first it almost seemed like a tast to get to the top of the block, but soon it was pretty easy. You will die a lot more than in most RPGs though, and I strongly suggest hoarding hommonculi. Nothing worrse than grinding nine floors of Tartarus, getting a bunch of cool shit and personae and then all of a sudden loosing it all because of an instant death spell.
Are you in the Velvet Room yet?