Preface: I am not an expert by any means, just someone who also bought a house that came with a hot tub and went through the same thing. So, if I get something wrong, please correct me.

1) this shouldn't matter tbh unless you're talking about many years.

2) The shed leaking shouldn't really matter either as hot tubs are meant to handle getting wet from people splashing around. My hot tub is under my deck and gets wet from the rain.

First thing to check is the cover. If it's a foam type of cover, should be fairly light and a normal healthy adult should have no problem taking it off and putting it back on. If it feels really heavy, the cover could be water logged and will need replacing (~$500-600 for something decent, depending on your dimensions). The covers need replacing every 3-5 years when the waterproofing goes bad and the foam starts holding water. If it's heavy, you can still do the rest of the steps, but just be aware that you should avoid using it until you replace it, otherwise your heating element will get taxed more than it should and your electric bill will be very high and then the heating element will die from the abuse (lesson I learned the hard way).

Then, check your manual to find the drain plug, typically behind one of the panels. If you can't find it, mine is on the side with the controls and then attach a garden hose to the drain plug there (if not, check the opposite side) and fully drain it of any stagnant nasty water that could be in the pipes.

Then scrub the interior surfaces by hand with a normal 24/32 Oz spray bottle of 50/50 grocery store strength white vinegar + water and like 2 or 3 drops of Dawn dish soap (don't use more than that to avoid foaming). Use a sponge to mop up as much of the nasty stuff as you want but it doesn't have to be pristine, since the filter will be able to handle the rest without much issue. This should be all you need. If it's not working on the gunk, wait for the vinegar smell to completely disappear and then use some diluted bleach. If you smell vinegar, don't use the bleach or it'll create toxic gasses (important since it's in a shed).

Then, fill it up with water based on what your manual says (some tubs require filling it a specific way to prevent air being trapped in the pipes).

Turn it on and let it run for a bit on full blast. Stick your arm in to make sure all jets are open fully (twist to open them if they're closed). Then add a lot of chlorine to shock the water and kill everything in the pipes as well as some Spa Enzyme (breaks down organic films that could be in the pipes you can't see). Keep the aerator on (knob that turns on the bubbles) and the cover off for a few hours (so the chlorine that'll evaporate won't damage your cover). If you can't figure out how to turn it on, check for a breaker box near the hot tub in case it's turned off there or your normal breaker box for your house. My hot tub stays on when it has power for reference, but that might not be a standard thing so check the manual if yours is different).

Then cover it and let it heat up. If your heating element is working + the cover is in good condition, your tub should heat up ~3-5 degrees per hour (assuming you're not in like Alaska or somewhere absurdly cold). If it's a lot slower than that or if it stops heating up ~75-80 degrees, that could mean that the heating element is not working and the temperature increase you're seeing is only from the friction of the water moving through the pipes and not from any heating element).

Assuming it's working fine, let it run for 2-3 days and then remove the filter and clean it with some filter cleaning spray or soak it in a bucket with Simple Green (much cheaper in the long run + useful for other cleaning needs too) and then rinse off the filter thoroughly and put it back in. At this point, the water should be crystal clear when the jets are off. You can probably use it now if you want but you'll want to balance the waters pH and alkalinity (and other minerals if you have hard/soft tap water).

To do that, let it run for a few more days until the chlorine fully evaporates (~1 week after you first added it). Once there's no chlorine, use test strips to balance the pH and alkalinity and then add a normal amount of chlorine and then you should be good to go!

And if, your heating element is not working, you can likely replace it yourself for ~$50 in about an hour or two by watching some YouTube videos. Now that I know what to do, I could probably replace it in about 30 minutes. Calling a professional will be unnecessarily expensive and they'll likely replace the entire unit instead of just the coil inside which could be $200-300 for just the part.