What happens with my contributions?

Your contributions are uploaded to the OpenStreetMap database and that data can be used for many purposes. Stuff like making map apps for routing, maps for background overlays, using the data to analyze trends, and much more.

How long before they're wrapped up into a map update?

Edits from StreetComplete are uploaded to the OSM database pretty much immediately. Afterwards it depends on whatever app you are using to pull that data in and process it. Some apps like OSMand have the ability to update within minutes, other apps like Organic Maps update monthly, and others update less often or on a different schedule.

Do other people have to solve the same quest as a double check?

No, once a quest on StreetComplete is uploaded there are no double checks. Other users can check for uploads from StreetComplete users but this is something separate and would be done after the data is uploaded. Though I believe StreetComplete may re-ask some quests after some time to confirm stuff like opening hours is still up to date.

This is off the top of my head so feel free to correct me if I missed something.

I recently ordered something to be delivered to a public locker, and apparently the locker service uses OpenStreetMaps to select which locker you want. I saw my recent contributions around the locker and it felt quite nice to have put my positive mark on the world even in a minuscule way.

Thy for your work :)

Now what?

Slread the word and get others to help us :)