Hey there! So I just read through the thread and I'll see what I can help out with.
First things first, for a 2 gallon tank with or without a filter it really does need to be changed twice a week, otherwise poop and other dissolved organic compounds will build up and start to hurt him, this is my initial worry for him. It's kind of like being locked in your bathroom for all week and you can use the toilet all you want but it doesn't flush except for maybe once a week, maybe every other week. Doesn't sound fun, huh?
So I want you to start off with actually doing three water changes a week. Scoop him out of the water with a cup, any cup will do as long as it has never been washed or anything so plastic is great for this. Then you can dump the old water and replace it with similar temp water from your tap, if you need to run a glass thermometer under the tap to get it at 78-80 degree's for your boy.
Then you need to acclimate him to the new water once you've used your water conditioner. So if you can, float him in his cup in the tank, then every 10-15 minutes add 2 tablespoons of new water to his cup so that he can get used to the new water chemistry. Then once it's been about 30-45 minutes you can gently release him out of the cup.
That's just to start helping for now. What it looks like is that he's just super stressed out right now, he most likely has blue iridescence but a red under color so when he's stressed out he's going to lose all that color and whatever his base color is, you'll see that.
From here it looks like his fins are really short, most likely from some fin rot from the water not being changed enough, but he might be a plakat as well? Can you show me a picture of his fins? Maybe a picture from when you first got him and now?
He also looks like he's got a white ring around his eye, do his eyes look swollen at all? If so, he's got a bacterial infection called Pop-eye, fairly easy to treat if you have the right kind of salt.
There are two different kinds of salt that we use; Aquarium salt (AQ salt) and Epsom Salt. They do two very different things and they are what I go to first before medications because they aren't as stressful on your boy. So AQ salt helps with things on the outside of a Betta like Fin rot, Ich and Velvet where Epsom salt helps with things on the inside; it reduces swelling, releases fluids like things to do with constipation and/or bloat and swim bladder issues.
So since he looks like he has some pop-eye issues going on, I want you to find some Epsom salt, you won't find it at the pet store isntead you find it at a pharamcy or walmart or somewhere like that. You are looking for 100% Magnesium Sulfate, no additives and no scent. At wal-mart it's like 89 cents or something, pretty cheap
which is good ^_^ so anyway, this is going to help with any fluid retention he's got going on and you'll use a 1 tsp per gallon.
So what you need to do is:
Change half of his water today.
When you go to put the new water in (so it'd be 1 gallon since he's in a 2 gallon tank) add your conditioner and if you can find Epsom salt, measure out 1 teaspoon.
Mix the salt in a different cup first, very important since when salt dissolves it actually makes and gives off heat which can burn your Betta!
So mix in a different cup, pour it into your 1 gallon that you have waiting.
Then scoop with a cup and gently pour into the Betta's tank so as not to stress him out.
After an hour, take out another 50% and do the same as above.
So now you have 2 teaspoons of Epsom salt in the tank to make the equation 1 tsp per gallon.
If you can't find Epsom salt today than just do a 100% water change to get him started. Also you should turn off the airstone, Betta's hate moving water. They come from Rice Paddies in Thailand where there is almost no water movement and lots of tall plants. That's why they actually have a labyrinth organ is to allow them to breathe air in low oxygen environments. So shutting off the airstone is going to help him feel better, you should also get him some ornaments when you get the chance since it's looking pretty bare in there. Betta's aren't afraid of big spaces, they're afraid of big open spaces. So filling that with some silk plants (Don't get plastic) and maybe some hides will really help him as well!
If he's having trouble reaching the surface of the tank than you should lower the water level about half way and do water changes according to 1 gallon instead of 2. So if you put the salt in, just use 1 teaspoon total if it's 1 gallon, does that make sense?
Sorry, I know this is a novel and all! If you have any questions on anything, please ask away and I'll try to clarify for you ^_^
EDIT: also, never go to any medications first, always use either one salt or another, do not use both AQ and Epsom at the same time. This is the least stressful for your fish unless you know they have something extremely serious, you don't want to stress them out more and antibiotics are very stressful.
Also **Note** do not use any salts for more than 14 days, especially AQ salt as it can cause liver/kidney failure which is fatal. So you can use more than 14 days, but they need at least 4-5 days rest before you start another regimen of salts.