For #2,
I guess you're using gm8/8.1 since you mention the yuuutu engine. When you die, use the sound_volume function on your current BGM to set the volume to zero, but let it keep playing in the background. Then you can sound_play your death music. On restart, use sound_volume again to turn the music back on. It's a little bit of a complex change if you're not familiar with how the engine tracks the currently playing BGM or where all the scripts are that handle these functions, but otherwise it's as simple as muting the BGM instead of outright stopping it. There's also a sound_fade function you could use if you want the BGM to fade out a little slower instead of an abrupt stop.
For #1 I haven't really looked at the new yuuutu since moving to studio, but if I were in your position I would get a copy of the old engine and compare the player physics code to see if I could spot the differences, particularly where the jump is performed.