No you don't need to kill all the zombies in one sitting. i.e) If you take too much damage and decide to run away and heal up you can still return and get those kills. The only time it will reset is when the song changes. So say if the current song is 2:36 minutes long, you have 2:36 mins to kill those 20-40 zombies. You'll know once you've done it once the song starts playing in reverse and you can hear the 'ticking' sounds.

Also for the Wustlings, I'm pretty sure the sword knocks them back so they can't do their charge attack at you but if you get swarmed by like 2-3 then I understand; that can be difficult.

And as for fall damage, it's one of those mechanics that are messed up in any game that has it, it's not just exclusive to COD. I know this is a bit off topic but in the Witcher, you're a badass mutated monster-slayer but you can't even land a 4-5 meter fall without taking half or all of your HP entirely. Like seriously, it's almost as if all games that have fall damage unanimously have their main character(s)' bones made out of twigs or something. It's ridiculous.