1) Let your insurance know. First, as has been said, if it turns into a fight with another insurance company you really want your in the loop. Second, at least our insurance, requires/wants you to file a report of anything that could become a claim (likely to blame you if you discover it and do nothing). Third, if you contact them later and you are already deep into the repairs they may not be able to determine the facts and it might not be a good thing for you.
2) Use whoever you want for assessment and repairs. Again our insurance has a list of approved contractors who can also to the estimate for how much the work will cost in addition to doing the work; otherwise our insurance will send out an assessor. It will be more work and time on your part if you have to deal with two parties and their schedules.
3) As you indicate you have a condo, you really need to determine the extent of your ownership. Some condos get funny on exterior walls and ceilings; sometimes you could own the entire wall/ceiling (jointly with the neighbor), half the wall/ceiling, just to the drywall, or just to the interior paint. Depending on the ownership, in addition to your insurance and her insurance the owners of the condo complex might also be involved.
4) If this were a fresh leak, I would advise against delay, the first 24-48 hours can be crucial to prevent the development of mold. Seeing as how this has been going on for "a while" the damage (and mold) have probably already been done, and since the source of moisture has been ID'ed and turned off, the damage is unlikely to get significantly worse between now and Monday. Still opening the windows and getting some fresh air into the condo would not be a bad idea.