Getting stuck in reverse and having pieces fall out of the front covers look like it's an internal transmission issue.
You're very likely to be dealing with damaged gears or shift forks but I couldn't precisely tell.
If that is the problem, the repair is usually beyond what most non-mechanics can do without specialized tools and knowledge of ATV transmissions.
It's a pretty intricate job to get everything back together correctly and ensure the shifting mechanism works as it should.
I would strongly suggest taking your Foreman to a qualified mechanic.
They'll be able to properly diagnose exactly what broke, replace the damaged parts, and get your Foreman shifting smoothly again!
It might be a bit of an investment, but it's likely cheaper than trying to guess and potentially causing more damage.