What is the fastest version of the game to play on? on top of that, is there any way to play an older version?

I lost my copies of most demo versions, but if you can get your hands on any of them you can play them.

Between 0.1 and 0.1.5 there have been a few changes to the damage formula, so there are definitely differences. I have no experience with Story mode so I can't say anything about that.

Puzzle Mode got a bit harder between 0.1.1 and 0.1.5 due to those changes to combat, but 0.1.5 came with quite a bit of additions to the whole mode, so you cannot compare 0.1.5 with anything below that anyway.

Exploration Mode is a funny one. 0.1.1 (iirc) is again the easiest one due to extra damage, but 0.1.0 had a bug where you could just walk straight into the observatory without getting all of the boxes (maybe you had to get both keys, don't quite remember). This makes 0.1.0 the fastest but also the hardest, as you go into the final challenge at level 3 with average gear. Another change important for Exploration is the removal of the "charge queue", where enemies in earlier versions could not charge attack at the same time and had to go one after the other. Again this makes the earlier versions easier and effectively faster.

With the release of 0.1.5 I have stuck with that version however. It's the most fun for both Puzzle and Exploration imo.