A very fun and interesting game you managed here. There are a few touches thrown back to certain old-school favorites of mine I like, and the overall package is just well realized.
In terms of game design this is well thought out. The stages have a natural sense of progression and are pretty easy to follow visually even with the intentionally limited color palette ("color" loosely used here; I'm an artist so black and white aren't really considered colors to me x3). Platforms are easy to distinguish and the collision detection for them is solid. I had no problems with suddenly falling through a platform I swore I clearly landed on.
The difficulty seems just about right. The game follows that sort of classic Nintendo model of gradual linear difficulty; there were no sudden spikes or sections where repeated deaths occurred. In fact-and this is just a personal thing-I felt the game was almost on the easy side, as I only died twice, both times due to enemy placement that cut off defensive options and seemed to call for sudden rush combat. Some others would consider these unfair difficulty spike scenarios, but given the relative easiness of the rest of the game for me, they were welcome and the penalty wasn't severe for dying to them or losing health. Generally speaking, my only real quip on the 'easiness' is w/ the bosses. I would have liked for them to be a bit more difficult, or the stages to be more difficult...one or the other.
Another issue I had was the game suddenly speeding up on the 2nd stage. It'd of been nice if it were that speed from the beginning, as it's a slight shock and takes a moment to adjust to, tho by the time one adjusts it is no longer an issue. I would have also liked more stages and a greater variety of missions per stage, such as defending a dame in the bar or a minigame of shooting alcohol glasses so a drunk can't get that last drink, to celebrate the theme more. There's also perhaps considering making the cover breakable by enemy fire after a certain number of damage is incurred. It also would be nice if there were more focus on utilizing the jump for specific challenging platforming sections. But I'm sure you're already considering such things ;)
Aside from that, I have no real issues with the game. The controls are very responsive and the jump feature nice. Any game w/ a double jump automatically reminds me of Super Ghouls 'n Ghosts, but unlike there you actually have control over your jump during the ascend and descend, which is a blessing. Momentum feels solid and exactly weighed, and the gun system is put to great effect and checked + balanced sensibly. Even the cover mechanic (well, moreso an extension of this type of game) is nicely done. The sprite work is charming and the music quite cool.
You have a solidly good thing going on here, and it's a very good first debut. W/ some more inventiveness and tightening up the challenge, I think you can create a grand slam. Best of luck to you on the follow-up(s)!