There is too much variation to give hard and fast rules about the roster questions. The trips seem to change month by month but, as a rough guide:
-When you go out flying, how many days you stay out of TYO?
Very rarely you'll have a there-and-back trip. Usually you'll pack up and be gone for 4ish days. There are blocks that are longer and shorter but I think 4 is about average, away from NRT. Quite often you'll find yourself in KIX, rather than NRT and/or staying in OKA before going 'foreign' again. As an example of the variability, I've just done an overnight, the next night flown back to NRT via OKA, arriving in the early morning, therefore being away from NRT for 2 nights. An upcoming trip has me leaving NRT, working through HKG and KIX a fair bit and then returning to NRT 6 days later.
Dont' worry though, the beer will still be there when you get back.
-How long are rest periods during layovers?
Vary between min rest (arrive HKG in the morning and leave that evening) to 36 hours ish. Usually there's enough time to go shopping (PVG), go for a rub down (BKK) or socialise with your mate (ALL).
-When back to TYO, how many days off in a row?
Usually you'll only get one blank day but can still get 2.5 days in NRT. For instance, arrive early morning on Mon, blank day on Tue, leave NRT in the evening of Wed. However, you can arrive and leave at all sorts of times, so that blank day can be as little as 36 hours in NRT. You can, of course, be on standby and this is often added to a blank day. You're not 'off'' but you're in NRT for 48 hours+. Also, you can have no days off in NRT; arrive the morning of Mon and then leave on Tues morning. There's no rule that says they have to give you days off in between all trips, so long as you have the required days off in a month.
-How many legs a day? Short, long legs?
Again, too many trips to put hard and fast rules. Sometimes you'll do one leg, even a short one like NRT-ICN. Sometimes that leg will be longer, NRT-BKK. However, with the developing network of OKA as our hub, a LOT of trips will now go via OKA, so most days (nights) will involve two legs, e.g. ICN-OKA-HND