A backup for sure, and even if everything should be fine, if you have some kind of vps feature for that, maybe a snapshot of the host 
, as surely there will be some database update to be done, might be easier than to restore from a backup if anything goes wrong (restore process that is interesting to practice by the way). Make a copy of the app.yml and write somewhere the SHA of the commit you’re on as an extra precaution ?
2.4 to 2.7, there are some postgresql updates if I’m not mistaken, maybe browse some of these Search results for 'postgresql update' - Discourse Meta as a warmup 
Yes, the container is stopped during the rebuild
Nope, not out of the box at least