My Photo library is in trash - how do I retrieve it without losing my photos?

The "Photos Library" is typically stored in the "Pictures" folder in your Home folder, not the Applications folder. See if you have better luck moving it to the Pictures folder, where the Photos app expects to find it.

You can also hold the OPTION key when launching Photos to tell the Photos app what library to use/open.

Again: the Photos Library does not go into the Apps Folder. It goes to the Pictures Folder. Put it there and try again.

Have you tried the "Put Back" command?

With the trash open, you can right-click or control-click on an item and you get the above menu. Of course, it may not always work...