Oh yeah, welcome to the world of 2 year old boys...it's an adventure!! :) The shoving and throwing are pretty typical. We went through a major hitting/shoving phase and I was constantly mortified, preparing to be mortified or hovering to hopefully prevent mortification. I think you should intervene in the shoving every time.
You don't need to cause a big fuss but start saying things like...we don't shover our friends, we need to share space and he gets a turn too, it's not okay to shove, we use nice hands, etc., etc. You can put him in time out, which I do think is an effective method, but at 2, it's still mostly a game, so if you go that route don't expect it to be a quick and/or miracle cure. Other moms do not like to see their kid shoved regardless of the reason, so I think intervention is key to happy park time. And even moms who maybe had a hitter or biter sometimes "forget" what it was like, so get your happy face on!
As far as the throwing...well if it's out of frustration I do think it needs to be addressed. Even a simple, it's not okay to throw when we are angry, will suffice. Something I say a lot is, It's okay to be angry, but it's not okay to ______ (throw, hit, call names, etc) and I do think he needs to clean up the stuff he threw. At 2 you will need to help him and that's fine, but he can't just throw stuff around, have a tantrum and then be able to just move on and not deal with the mess.
Also start encouraging him to ask for help. You can even teach him the sign for help. Maybe if he had that in his repertoire it would be easier for him to communicate. My son was a late talker and we only did a few signs but they worked wonders!!
As far as the nose...I'd ignore it. It's not going to make it go away, but you have enough other stuff to work on before tackling the nose! ;) Just keep tissues handy and hand him one every time you can.