Photos App on Mac not working

It's not the Photos app that gets damaged it's the library. If on an external drive a Photos library MUST be located on a local drive formatted either APFS or OS X Extended (journaled) and be directly connected to the Mac via cable.

Have you tried launching the old library (once it's back on your boot drive in the Pictures folder) with the Control+Command keys held down to repair the library?

Do you have a backup of the library from before you moved it to the Dropbox? You could start with that if all else fails.

When you move a Photos Library to a synced folder on Dropbox, Google Drive or any other folder syncing with a cloud service, the syncing can damage the internal links and cause a loss of photos. If you wait too long to rescue the library, it may be too late and Photos will not be able to repair the damage done by the syncing. In that case you have to recover your Photos Library from your backups.

The only safe way to save storage by keeping your photos in the Cloud is iCloud Photos. Turn on "Optimize Mac Storage" in the Photos > Settings on your Mac. Then Photos will try to keep the local copy of your Photos Library small.

OK-- we have to figure out where the problem is, so you need to try these things:

• Restart the computer (of course)
• Re-start in Safe Mode. This bypasses certain potentially disruptive processes. Safe Mode is different for different computers, so see this: 

     Start up your Mac in safe mode - Apple Support 

• Make a new user and see if the same problem recurs with it
• Try to add some pictures to a new Library (without declaring it the System Library) and see if it works with new pictures.

The idea is to find out if the problem is with the Library, with your account, or with caches and login items, and stuff like that. One of these steps may fix the problem, but their failure to work also helps figure out what's happening.

Do you use iCloud to synchronize the pictures on your Mac and on your iPhone? If so, have you looked at iCloud.com to see if all your pictures are there? If they are, then the best thing to do in this case is, hold on, make a new empty Library.

To make a new Library for the Mac, close Photos, option click the Photos icon, and choose Create New from the Library dialog. 

A new, empty Library will open. Then, on the Mac in Photos>Settings>General, you must make this new, empty Library your System Library. 

In this picture "Use as your System Library" is grayed out, because it already is the system library. Only the System Library can connect to iCloud.

Go to Settings>iCloud and connect this new Library to iCloud.

Set "Download Originals," if you have the storage space available on your Mac. If not, you can check "Optimize."

Then wait. Wait some more, and eventually this new Library will fill with full sized images from iCloud.

Make sure this Library has all your pictures, then trash the old Library.

Let us know if you use iCloud and so if this works for you...

[ Well, just like markwmsn said, as he hit enter a bit before me...]

If your iPhone has iCloud Photos enabled, you can create a new library on your Mac, make it the System Photo Library, and enabled it for iCloud Photos. If your Mac is tight on space, you probably want to choose the Optimize Mac Storage option. Your Mac should start downloading all your photos from iCloud (in reduced resolution if the Optimize option is selected).

Photos is warning you that, if you have Optimize Storage turned on for your previous Library, then many of the pictures in that Library will be smaller versions (to save storage) and not worth saving. Luckily for you, all your pictures are in iCloud, so you don't need that old Library.

I'm not sure why Photos is crashing. I think the first thing I would try is to put any already existing Libraries in the Trash (you don't have to empty it, yet, but you may have to OK the transfer) and then open Photos-- it should make a brand new Photos Library all on its own. and hopefully it will work.