I accidentally deleted a whole bunch of stuff from App Inventor. I would like to know whether I can recover them, and if I can, how to.
it depends.
If you regularly create a backup aia regularly and save while you code, you never have to worry as you will always have a recent back up.
That means doing
to
- save project as .... to a name you choose
- create a check point ... similar but save a copy
- export selected project (aia) to my computer
Also if you are in the process of coding Blocks
you can click on an empty area of the Blocks Screen to show this screen and
Undo or Redo to get back a block you just deleted or to delete again.
Also, when a message pops up and asks if you want to delete you always can select Cancel or No instead of quickly reading without thinking and say yes Don't lose code accidentally. There are safeguards. 
More:
and other messages similar to it.