Your plan is insane. You are trying to do far too much in your allotted time (and you will probably go way over budget.)
Let me keep it simple for you. 'Hitting up', as you put it, 10 countries or cities in 30 days (ignoring the fact that you'll be jet lagged at the beginning) leaves you with fewer than three days per place. Since you need at least one-half day (and in many cases travel the travel will take almost a full day) between places, you're down to fewer than two days per place. So you just got somewhere and are already planning the departure. Your quality time in each place will be far less than this since, if you are on a budget, you'll likely have to choose public transit from an airport to a city, rather than a taxi. I expect that you'll arrive in a city on the evening of day one, have possibly day two to look around, and have to leave again early on day three to 'hit up' the next place. Sounds like a complete nightmare to me.
You should also consider that, whilst taking the train has distinct advantages (scenery, going from city centre to city centre, etc) it is generally slower. LCC airlines can also be cheap, but often land at obscure airports far from where you actually want to be, and fly at unsocial hours. $50 per night accommodation is possible in theory, but you're looking at YMCA/hostel quality.
I think you should reevaluate where you really want to stay and what you want to see, and choose no more than about 5/6 places.