>>>I can get a nice enough B&B in Rome or Florence in winter for $50 <<<

What's the name of these nice B&B's. That's only 35€ per night. Rome is a major city and doesn't have much of an off season (perhaps mid-January or August).

Italy will be hot so you should look for places with AC.

Yes, you can get a bed in a dormitory style room at hostels for $US 20 per night. There aren't too many at that price and they might not be central, but they exist. If you can splurge and go up to $US 25 there are more options. At $30 US you should be able to find something even in Rome, even something central.

Besure to scrutinize Rome 'ostellos' around the train station - I ended up in one that was a real dirty flop house - beds hardly fit for sleeping in - doors that would not lock, etc. Do not book any Rome station hostel blindly IMO - especially if there is a low low price.

if you plan to visit Munich don`t miss the visit of NEUSCHWANSTEIN

you can get to the castle by train in two hours and for a budget of 34Eur =22 (Transportation)+ 12 (castleticket)

check how fantastic is the castle
http://www.new-swan-stone.eu/p/hotel-hostel.html