Absolutely it's fixable.
The L and H screw should never need to be touched normally - they act as stops to prevent the chain going too far and jumping off the cassette.
The cassette will end up in the same place on wheel reinstallation, and if not then something has changed. Check that the axle is firmly in the dropouts, and nothing's hanging up on brakes or mudguards. Also check if washers are on your axle and have ended up changing sides of the dropout.
As for getting your bike back in order
FIRST UP there are a lot of chompy moving parts that want to eat your fingers and clothes and hair, etc. Be mindful. Also don't get the wheel going too fast.
Try and get the back wheel off the ground. A workstand is nice, but rope or old inner tubes can be used to hang it up too.
Set your right-hand gear shifter into the hardest gear while slow-pedalling with your hand.
Adjust the H screw until the chain sits on the smallest cog while slow-hand pedalling.
Then do the same for the L screw, shifting to the "easiest" gear. Be careful not to jump the chain completely over the cassette, this can cause damage at speed.
Tweak the L limit screw until the chain is just making it up to the largest cog.
At this point the limit screws should be right, and never need touching again. Then move on to indexing.
Use your left hand to turn the black plastic barrel nut that is on the rear derailleur, around where the cable enters. Tightening will effectively shorten the outer housing and moves the chain toward the smallest/hardest cog.
Likewise, backing out the barrel nut makes the outer longer essentially tightening the cable, like you're pulling on it with the shifter lever making it move toward the easier gear with the most teeth.
You can test by shifting with your right hand, and use your left hand on the right pedal. There may be some compromise between shifting once per click up compared to down. Some iterations will be needed too.
Finally, your last resort is that if all this sounds too hard, most bike shops offer a tweak-up service on bikes they sold. All new bikes have a setting-in period where the cables develop a set, and everything beds in. This may be free, check with the bike shop.
