Is there some sort of app that will let me, preferably toggleable with a keyboard shortcut, hold down the left mouse button? (Not click fast, just hold down).
The accepted answer from 2012 is obsolete. Use the following for Mac OS X:
Mac OS X
System Preferences > Accessibility > Mouse & Trackpad > Enable Mouse Keys
Mouse Keys allows the mouse pointer to be controlled using the keyboard number pad.
What if I don't have a "keyboard number pad"?
If you use a laptop that does not have a numeric keypad, you can still the Mouse Keys feature, but you will need to press the Num Lock key first. Once you have turned on the Num Lock, use the keys with the small numbers on them to control the pointer. The I key will perform a mouse click.
What if I don't have a "Num Lock"?
I give up. Search the Apple Store or Google for a 3rd part app.