Macbook Pro slow/not booting after battery replacement

(Macbook Pro Retina 13” Late 2013 2.4GHz i5 4GB ram)

in early 2016 i noticed my battery was due for a replacement, but i ignored it since i never moved it anywhere anyways. So in the middle of 2016 it was completely drained and couldn’t hold a charge and wasn’t even recognized by the system when I checked the system analysis thing. So from them till recently it never booted normally, had to sort of jump start it— hold power for a few seconds, plug it in, take my finger off power for half a second and push down again. It worked for me but immediately, the fans would kick in at full speed until I unplugged the charger, which just shut down the whole thing. Using my macbook in this way was working for me although with the slight inconvenience of slightly reduced speed. Just slightly. But still noticeable. like logging in would take about a minute instead of a few seconds and graphics would lag “prematurely”.

Last week I finally dropped it off at the apple store and they replaced my battery but the guy told me it wasnt booting properly. He told me to boot it by holding power+ctrl+alt+shift(shift, i think.) because otherwise it would just chime then turn off and the power button was "not working" on my 2nd visit (picked it up on the 3rd.). But i found it’s less stressful to just hold power again after it turns off. Anyways when i get to the login screen it takes about 3 minutes to log in, the bar gets stuck after like 85%.

Now my macbook is drastically slower, it takes like 4 seconds for the volume box to respond to the keystroke and the same goes for the trackpad backlight and the display brightness. Videos can barely play if you’re doing anything — ANYTHING other than watching them. Played a couple games and they’re all running so much worse than they did with the terrible speed i had before. I updated from el capitan to Mojave to see if it would make any difference but it didn’t. Cleared about 30 gigs of space too and that didn’t do anything. I have a 2014 macbook air with a battery approaching the level of bad that my pro was once at and it still runs faster than it, and it’s in Yosemite.

Does this have anything to do with the battery? I can still easily crank out 7+ hours of life so i assume it’s the right battery. Still takes as long to charge to full as I remember too. I have to admit i probably skipped tens of updates over the years but i know my SMC is up to date. i’ve reset the SMC, PRAM, NVRAM, safe booted and everything, still the same problems. I’ve had the idea of pulling up so many programs that the fans kick in and everything runs at full capacity or whatever “to get it back into the swing of things” but it sounds dumb.

Im assuming if i could get it to boot on the first try it’d be in a normal state and run properly?