The top picture does seem to show what might be the beginnings of fin rot, though it doesn't seem clear in the other pitures. Fin rot often shows as a blackening to the fin edges, and they will begin to disintegrate leaving the rays protruding ( these often dissappear too )
Going without a filter is not a good idea at all. Espescially if the fish is ill. A 3 gallon is too small to filter properly, which means that you will need to do daily water changes of almost 100% on the tank. If your fish is ill, this kind of stress is not good for him ( it's not good for them even when in good health ) it will also remove a lot of any medicine you use in the tank every time you do it. Kind of counterproductive to any treatment really.
You describe a shivering movement when he swims. Does he flick or rub himself along the plants or bottom like he has an itch? If not then that's good, but if he does, then this is a sign of irritated skin and slime. Things like that are usually caused by water quality problems, and since you have no filter, your risk of such problems is higher if for any reason you don't or can't keep up with the daily water changes.
The best thing to do is treat him with a half dose of waterlife Myaxzin for a week or two, to clear any fin rot, and look around for a bigger tank ( 5 gals + ) with a gentle filter like a sponge filter, or one that is built into the lid of the tank. He will have more room, and the greater volume of water will mean his waste is far more diluted. This, along with filtration will result in much cleaner water and less risk of future health problems for him.
He may have contracted this possible fin rot from the place you got him, and the stress of moving will not have helped. Medicine and very clean water will help to heal his fins , and they will grow back.