Eye problems, not eating, maybe impacted? Also not moving back legs normally.

there are several things wrong with your set up.
when you say a heat pad on one side do you mean under the tank or literally on the side of the tank?
if it is infact on the side you need to move it onto the floor of the tank as geckos absorb heat through their bellies.
you have not listed vits or supplements, dose this mean you are not using any?
if this is the case its extremely vital you start supplementing with a mix that contains calcium and vitamin D3. (you can leave a lid full of pure calcium in the tank and dust his food with the mix that contains vit D3).
a lack of vitamins and mineral could be the cause of your leg issues and lack of heat the food and impaction problem.

The heat pad is on the bottom of the tank, on the side with no hides. Also, I just checked it out and it doesn't seem to be working all that well. I stuck my home thermometer in the tank (don't tell anyone) and it read 77 degrees. The light turns off at night, so the temp probably gets to around 69 which is room temperature. Would replacing the bulb with a higher wattage one help? I have not been supplementing his diet in any way.

Since he is not eating, what is the best way to give him calcium powder?

Oh also, I recall reading somewhere that the stuff in his eye could come out next time he sheds, is that a safe way to deal with that problem?

having a light on the tank is purely for your benefit (to se the gecko ect), the lights are not useful to geckos atall as they absorb heat through their bellies.
if the mat is not working i would advise replacing it asap as its vital for the geckos survival. the temp is a little low on the hot side, during the day it should be around 85. temps falling at night is normal.
sandwich the mat between some tiles and try placing a medium sized hide on top of the UTH, as this will offer a spot that is much hotter. its good to offer spots of varying temps to allow the geckos to thermoregulate properly.

It is important that you start supplementing as i have already stated. if hes still not eating you could try a slurry mix, there is one on here somewhere by golden gate geckos. or alternately you can just blitz up a mix of bugs with some minerals and vitamins and a tiny bit of organic chicken or pumpkin baby food (thickens the mix up). use a small syringe to place small drops on the lips. the gecko will then lick it off its lips.

the mixture can be put into an ice cube tray and frozen so it will last longer, just get one out every day befor you feed.

if there is skin on the eyelids it should come off next shed however there should not be any skin on the eyeball itself. if this is the case he might need to see a vet to check that it is skin and not another eye condition.

I gave him this suppliment stuff today by wiping some onto his lips for him to lick off. I also used a q tip to put water on his lips for him to lick off.

Here is a full body shot, he is sitting in some warm water.

This is his left eye, the bad one. The whiteish stuff is basically gone. However it is rather cloudy and he still likes to keep it closed. I put a drop of saline solution in there.

Here is his right eye for the sake of comparison.

Is what I am doing correct? Thanks a lot for the help.

he looks in reasonably good nick, his tail is a tad thin though.
supplementing is a good start but you need to make sure your getting food into him too. blitz up some bugs (in a food blender) or mush them up by hand if you're not squeamish with a small amount of that calcium mix.

his eye lids looks quite swollen. i iv recently had a male with a similar eye problem (it needed surgery), anyway the saline solution i was bathing his eye with actually made it worse. so id say try just using tepid water instead of the saline for a while and see if that makes a difference. I just used a fresh packet of cotton wool (bought especially for him so there was no chance of contamination), dipped it a pot of tepid water and used the cotton wool to gently drip the water into the eye (to flush out any thing in the eye, same as you would do to your self if you got any thing in your eye) then I gently wipe over the closed eye its self.

Hes cute he looks like my sausage.

I agree with sausage that your gecko looks like he's a pretty "safe" weight so any issues he's confronting should be able to be dealt with before he's in serious jeopardy. Sometimes shedding issues can develop from a vitamin deficiency. You could try soaking him in a gecko sauna, or in other words a closed tupperware with about a 1/2 inch of 95 degree water for 10-15 minutes every 24-48 hours for a few days which should help temporarily with any shedding issues. I think your best bet would be a trip to the vet. If you haven't had him on adequate heat and he hasn't been getting the vitamins he needs in addition to the calcium the vet may be able to diagnose where a deficiency or problem lies and give you some supplements that would help him the most quickly and best. Good luck and keep us posted on how he's doing!!

Is there any good way to increase the heat in his tank while I wait for the opportunity to get a new pad?

Also, for planning purposes, about how much will a vet visit cost, assuming he just needs antibiotic drops?

try placing a hide directly on the heat mat, that will create a hotter area.
all vets will vary tbh.

I will finally be seeing a vet tomorrow, would have gone sooner but my grandmother got very sick.

Also, a new heat pad got put in last week, he seems to be enjoying it!

Update from the vet visit:

He has infections in both eyes, right eye is quite mild, left is moderate.
Somewhat dehydrated
Getting rather skinny (52 g)
Calcium deficiency

Treatment:

Anti-Biotic eye drops
Subcutaneous fluid injection
Daily warm water baths
Oral calcium suppliment
Carnivor Care suppliment until he starts taking interest in mealworms again.

Prognosis:
Full recovery