This does have a retro inspired feel to it. The sound and graphics feel appropriate to the retro feel. The player response to the controls is decent. Not great, but it does fit the feel of the game.
I'm not really a fan of using Z and X for hand positioning. I found that space worked for jump, so I used that, though key remapping, or just some alternative input options, would be nice. I also found that R restarts the level, and P pauses the game, though without any notification. I was testing buttons when there was nothing else on the screen and after pressing a bunch suddenly nothing was working. P made the most sense, so I pressed it again and moved to see an enemy before trying it again to confirm it paused. I recommend adding these to your controls list on the web page, as well as adding something to indicate when the game is paused. Since you are listing controls, it would be good to include mention that arrows are used to move. I'd also add something to the start screen saying to press one of the buttons to continue. It could be a loading screen, so I waited a bit to see if anything happened before clicking and pressing buttons.
I made it to the end, but did not figure out how to get some of the items, or to the door above your entry to the second area. I didn't spend too much time on them, but did try bombing everything nearby.
The level design felt fine. The difficulty was reasonable throughout. The ceiling enemies were annoying, but in a way that works for the game. For the ones shooting out your path I just ran through the area, rather than trying to take them out.
There was a spot where it looked like an enemy just appearing on screen fell onto the spikes and died immediately. This was in the second zone.
Overall, this feels like a good start.