Hi and Welcome.
Coping with a poly partner when you're mono (which you seem to be) is in my opinion a process that you have to go through. If you want to have a look at my process the link is in my signature. Everyone's process is of course unique but there are commonalities such as dealing with jealousy and insecurity. Monolicious has also written some really good stuff about this recently on here that shouldn't be hard to find.
The thing with jealousy is that it will highlight problems in your own relationship. You say you and the other guy are both feeling short changed in terms of time and that your wife is a student and working. It sounds to me as if this is where you should start. It would be difficult for anyone to successfully maintain all this, but add to that the fact that you're struggling to cope with the first other male in your marriage and he is depressed and drinking, and it all sounds too much to me.
Let her let him go and but don't take all the responsibility for it on your shoulders. Discuss it in terms of feasibility in the current context of your life. Let her know that you're willing to explore being able to cope with her pursuing relationships with other guys but that maybe this guy at this time isn't a good fit.
Of course she'll be sad, but her life is full and if you aren't beating yourself up over it you will be in a better position to support her through it.
As far as him self-harming, that sounds like emotional manipulation to me and more reason to leave well alone.
Hope that helps.