Saved Snippets is very helpful; pre-empting frequent/common questions, both with an FAQ in your policies (there's a policies tab on every listing) and with carefully constructed listings helps.
We get 50-100 convos a day and I (Sara) answer probably 95% or more of them. It is a grind. Sometimes I get behind by a day - never good! Because they're still there the next day.
I find allotting a specific time every day to sit down and answer convos helps. Usually for me this is the very last thing I do every day before bed (I am doing it now, in fact!). If I ever get woefully behind on convos I put up an announcement to that effect but for the most part I can get through 50-100 convos in a couple of hours, so it's more efficient to just go through and answer them.
I do triage them throughout the day and try to scan for ones that absolutely need to be answered immediately, but those are pretty rare. Most inquiries/problems/requests can wait a few hours.
You can put in your policies that you only answer convos once a day, but I don't know that it makes much difference - very few people read the policies, or the entire listing description, or your shop announcement.
The main thing is to cut down on the volume of convos by pre-empting some of the common ones, and to use saved snippets.
