First and foremost you have to understand the differences between a keyboard and a piano. The most significant difference to me is the key touch (the way how the key feels when it is pressed). Piano keys are usually a solid block as compared to keyboard keys which is pretty "hollow". I suggest you go to a music store and try out both keyboard and piano to get what I'm trying to say.

Do note that not all keyboards are 61 keys. There are 41 keys, 88 keys etc.

You should be able to play quite a number of songs, provided they don't require a full 88 key length (most songs don't btw). But if you are to transition from playing keyboard to piano, you will need some time to get used to your new instrument. I'm not saying that you must get a piano to start playing, but if you are looking  for long term, a piano would be better. However, many players do start by practicing on a keyboard first, then when they feel they are more competent and wants to get serious in playing, only they'll buy a piano.

Ahh another interesting topic to discuss about. Well, digital piano may not sound or feel 100% like an acoustic piano but they're much more versatile compared to acoustic piano. But my biggest concern with digital piano is that they tend to get outdated in a few years time and electronic parts are more likely to break down as time goes by