shouldn't be a problem as long as you can provide it with a static public IP address (probably have to buy one somewhere, domestic ISPs/providers usually do not give you one for free)

That said, you will probably need a good power supply in order to have a high uptime, and microSD cards are probably not meant for such constant wear-and-tear (the master node writes the whole blockchain to disk). They are also notorious for corrupting your data on a bad system shutdown. So you might have to connect a physical hard disk. I have never looked into this though, just google for some DASH raspberry master node tutorials and you'll be fine.

Unless you are really enthusiastic about using a raspberry just for the coolness factor (and low electricity cost), paying for a small VPS will save you a lot of time and headache and thereby money.

Another question: is there a way to see how many masternodes currently exist in total?