Yes it can do. Granted it's rare in non-mountainous areas but it can and does happen.
You will find snow in the mountainous regions of countries such as Switzerland or Austria, and Italy. If you want the resorts that are open then you can get some ideas here for example http://www.onthesnow.co.uk/news/p/1133/a...
For snow in non-mountainous regions then it's unusual for it to snow in e.g. Spain, Portugual etc but can happen in northern Scandinavia Norway, Finland and Iceland where snow can appear from October onwards, so it's very possible.
If you want snow then the further north or the higher in altitude you go, the more chance you have of getting snow in Europe.
Definitely
Yes
yes. but its most likely in eastern Europe or scandinavia
Depends on how fast global climate change sets in
