stereo cartridge on mono record

yes you are fine.

That will depend on the age of the mono record. Pre 60's mono maybe not good, as the needle profile if different. If 60's and later should be fine. The album cover could say ok for mono or stereo needle. Some record companies have two numbers like these CL or CM then numbers on the album cover for either mono or stereo, these should also be ok.

The follow link shows how to to wire a stereo cart for mono playback, you might want to take a look at it.

http://pagesperso-orange.fr/jlf/erepro2.htm

A stereo cartridge is sensitive to vertical modulation of the groove. A mono record has only horizontal groove modulation, and whatever vertical movement occurs (and this includes turntable rumble) is pure noise, and you don't want it. It is essential to remove sensitivity to vertical modulation by jumpering the two channels together, or using the MONO switch on your preamp (if it has one).

Improved results may also come from using a different (larger) stylus, and these are available for some stereo cartridges.

you are fine fine fine