Not sure why you can't see when you last played on a town (for your Birth date day) - not showable in activity log?
Well I play many towns, I'm unwilling to say how many, apart from saying I'm at the very max that I can deal with, if not over). But each one means a lot to me, they all have town/character names that link back to special happy times in the past. I could not ever bring myself to delete one.
They are all on cart, all on their own 3ds (I'm an adult, its my choice to buy, I buy a unit if on offer & I won't be buying any more, yes a luxury, but its my main/almost only form of relaxation).
I never try and keep up with anything, these games are my relaxation, stress free time. They are all still in 2013 (well one is near to coming up to 2014) - I set them in 2013 even tho the last one I bought was last year. No necessity to do this, I just did. (2013 is the year my father died, maybe that has something to do with it).
There are no issues with being on a different date, system updates (for the unit) come down OK, DLC (when we had it, I'm in UK so I've not had any for a long time) comes down fine, whatever date you are on.
Streetpass works OK, when or if I get it, that's not down to the date its set on tho.
You can visit other towns (to sell turnips etc) set on any other date, past or future. [Turnips spoil based on the date you buy - I always buy in my own town, going to a town on a different date to sell, has no effect].
Unsure if how you change time has any impact - I don't think it has.
Since acww days (where it was a necessity) I change time on the ds unit in system settings and then load the game.
I keep both times in sync as I only play acnl, no other time related games. I find it simpler if only because it gives me a 2nd chance of seeing the time I save (if I forget to note ingame I can see it on the unit itself). I make a written note of the time I save.
I only move forward ingame, change the system clock back to the time I want, after the time I last saved.
No stress, lovey easy game to play. But it is a game and does not take the place of real life, it gets put on hold for real life issues or if just don't feel like playing.