Arrow keys or A and D to run, and Up, W or Space to jump. On a phone or tablet three pads sit under the room: left, right and a wider jump button. Holding jump longer makes you jump higher.

Press into the wall and jump. The direction you are holding decides what the jump does: hold toward the wall and you climb it, staying close enough to catch it again and go higher, so a single wall is enough. Let go of the wall first and the same button leaps you away from it instead, which is what you want for crossing a shaft. Sliding down a wall also slows your fall, giving you time to line the next one up.

Probably not the timing. Leap gives you a short window to jump after stepping off a ledge, and remembers a jump pressed just before you land. If a jump is failing it is usually distance rather than timing: try running at the gap from further back so you are at full speed when you leave the ground.

Ground that gives way. It holds for a moment after you land on it and then falls, so it is a step and never a place to stand. They come back when you respawn, so a failed attempt never leaves a room impossible.

No, and that is on purpose. Every room in the game has been verified as completable before shipping, so being stuck always means there is a route you have not found rather than a broken level. The shop only sells skins — nothing in it makes a jump easier.

Yes, automatically in your browser. You come back to the room you reached, with your coins and skins intact.