This is an easy fix albeit I will caution you, what I refer to as 'chain jumping' is one of the more dangerous scenarios when riding a bicycle. Each time it jumps, from pushing hard on the pedal, the worse it becomes.
For the moderator, my experience comes from building over 400 bicycles, 2/3 of these vintage mountain bike conversions to road bike. Nearly all bicycles I received I later gave away free, yes the Cannondales also.
I took a look at your first video which showed me the chain jumping on the same link each and everytime. This is simply a bent link, or rusted, or too tight. Lube it, check alignment closely, is it the same width as the other links? If it is okay your fix is NOT the front derailluer but rather, the barrel adjuster on the rear derailluer. Only turn this 1/2 turn per test. It will solve your issue. Having said that, please check this prior to doing anything. Flip the bike over, get on one knee behind the bike and look at the alignment of the rear derailluer. Are the cogs - wheels on derailluer -straight? That is to say at both 180 degree angles. Straight up and down with each other and straight - lined up with chain travelling to front derailleur. You can get away with bending them by hand but he careful. At this point also check for looseness or typically with your issue, over tightening of one, usually the lower, cog/wheel. The sleeve inside the wheel allows travel/spinning so clean them! WD40 is a good cleaning agent however, it is not a lubricant and will attract dust.
Finally, I see you've purchased and installed a new chain? Is it exactly the same length as the original? If so the jumping on the rear cassette should have stopped. However, the chain falling off the front cassette is worrisome. You've either too long of a chain or, you should start by readjusting the rear derailleur from scratch.
Want to prove my response? Go to this link for the complete guide and trust me, for mountain biking the front derailleur is an engineering overkill (unneeded).
Good luck!
https://www.bicycling.com/repair/a33969991/how-to-adjust-rear-derailleur/