Then how can I get over the large gap in that level?

To cross that gap, jump and immediately shoot to the left, then hold right.

Cool game. What am I supposed to do here?

To get to the other side, jump straight up, then when you are touching the bucket of slime, shoot right then immediately shoot left so you don't fall in the spikes. Then, stick to the wall, and do this: walljump (no angle), IMMEDIATELY shoot right (so quickly that the bucket refills your slime), and then shoot down to get enough height.