A good game you made there. Most errors can be corrected with a little fine tuning.
I like the jump mechanics very much. They are fast and don't feel floaty. By holding down the spacebar, you jump higher, which is really nice. The only thing that feels weird about your jumps is the movement speed to the sides. I would tune that down so that it matches the ground movement speed.
The climbing part works pretty good. THe only thing I would add is some kind of animation, which shows that the player is holding on to the wall.
The graphics are pleasant to look at. But I think they can be enhanced by a background and frame by frame walking animation. The circle around the bombs, however, feels rather off, since it has such a low resolution.
I also like the idea of objects that show fake spikes. But if you decide to use bombs, I would tend to actually blow up the spikes instead of just showing through a mask where the real spikes are.
Another thing I noticed is that the graphics blur when they move fast. I don't know if this is a problem on my side or if you added an effect that causes this. Either way, I prefer crisp graphics.
Two last things: The player shouldn't be able to move after death and you can cancel the loading process if you run into a door again.
Please don't feel too bad after my review. I just prefer to give more detailed feedback.