Question 2: Can you get the achievement for beating the higher difficulties if you go through the game again with all your items/upgrades? 
 
Question 3: How do you unlock the hardest difficulty? By beating the hardest available, or by beating it at all, even on easy? 
 
I bought this game week 1 because I heard such good things and was pretty hyped to begin with, but decided to stop after almost an hour and wait to find the answer to these questions so I could know how I wanted to play it. I hope I can just play it on normal/easy and enjoy myself, and then beat it on the very hardest in a new game plus. That would be fun. But I don't want to play it on normal, only to find I have to play it again on hard to unlock Harderer. Twice could be fun. Three times not so much. 
 
-Thanks!

you can go back to any level at any time with all your stuff

You unlock Harder after beating the game and difficulty can be changed anytime via the options menu.  There are a lot of items in early levels that require skills/relics that you'll get later in the game so replaying levels is encouraged if you want to collect everything.

You can just go back to each individual level and beat it on the harder difficulties to gain the achievements, you don't have to do a straight run through to get the achievements,