I'm glad you are enjoying this game, I put a lot of work into it!
So the saving issue... I haven't done much testing on the itch.io site, so there may be an issue there, but typically the game will not save unless you prestige. It was designed to be a rogue lite type game with progression (which is not normal in roguelike/lites). In order to progress you need to eat a Humble Pie, which will save all of your prestige bonuses and restart you. So you will start at an advanced scene, with bonus xp and the elemental stat bonuses from the other presitige items (Toughen Muffin, Pricey Spice, Breezy Cheese, Funny Hunny, Dandy Candy and Vitali-Tea). You'll need cookies enabled, your browser may not prompt you about cookies until you attempt to save.
For the weapon statistics, I noticed that same issue the last time I played it, currently looking into getting the inspect button to work with gear. It was an oversight and it was intended to work with those.
I registered over on audiogames.net and posted in the introduction thread. Most of the comments there have to do with the saving, which it looks like a few people have already guessed that it is a prestige function. So hope these posts clear things up for you all!