They don’t disappear from your sent offers list. The requests disappear from the active list if you’re a level 0 seller and enough other level 0 sellers have sent an offer (10?). A buyer can stop (or pause I think) a request or it can automatically stop (not be shown in any active list, including level 1 seller or above) after 30 days, though I think the buyer can resubmit then so occasionally I think there can be requests > 30 days old (but the vast majority won’t be >30 days old).
They don’t automatically disappear from the list when someone has found the work.
Probably normally within a few days I think (eg. within 3-4 days I’d guess normally). Maybe the highest chance is on the same day the request is posted. eg. if they message you and you can show you can do the job they want.
You could search the buyer help page about posting a buyer request to see how that part of it works.