Hey there mamapugs. :) You've been getting some great advice so far, and from what I've read it sounds like your guy is doing much much better! Hes certainly a tough little guy, and very lucky to have someone who cares so much about him like you!
I just have a couple odd bits to contribute....
Rather then try and mess with establishing a cycle in your tank, I would recommend just removing the filter all together. Trying to establish a stable cycle in a tank under 5 gallons is very tricky and really isn't something you want to try for if you don't have previous experience cycling aquariums. With the filter running and a lack of regular water changes, this could have been the cause of, or at least contribute to if you have some ammonia in your tap(wasn't sure if you had tested that or not, have you?), the high ammonia levels as your tank was probably trying to start and go through a cycle.
At this point, I would personally suggest a full 100% water change in your tank. Take everything out, give everything a quick rinse, then set up the tank again with all fresh water and without the filter. Add your conditioner, then slowly acclimate your betta back into the fresh water. It sounds like your little guy has been through a lot of meds and treatments recently, at this point since he seems to be acting well and his velvet has cleared up, I think your best bet is just to keep him in warm, clean water and a little stress coat for now and see how he does. Warm, clean water can do quite a bit on its own, more then you might think. ;)
If you have no ammonia in your tap, a 100% change should get all that ammonia out and, since he is doing better, you could probably go down to 2 50% and 1 100% water change per week to maintain the water quality and just let him rest and heal and recover.
If you do have ammonia in your tap, you may want to, for the long term, consider a larger aquarium that you can establish and maintain a proper cycle in to deal with the ammonia.
I would also suggest picking up a different kind of conditioner by the way. I'm quite surprised no one seems to have suggested this(or perhaps I missed it?), but Seachem Prime will lock and detoxify ammonia for 24-28 hours as well as get out all kinds of other stuff that a conditioner like Stress Coat/Zyme won't, such as heavy metals. If you have Ammonia in your tap water this is going to be especially helpful.
Stress Coat/Zyme is still fine to use, its great in helping to promote healing and a healthy slime coat so it should be helpful in healing up those ammonia burns, but I wouldn't trust it as a sole conditioner to get all the nasties out.
And lets see....as a side note about both the rings around the eyes and the poo...
Poo can sometimes hang around for a little while, I see it happen every once in a while with my guys. So long as it appears to be a normal color(somewhere in the brownish range) and isn't long or white or stringy, and since it sounds like it did eventually fall off, I'd say thats fairly normal and nothing to be concerned about.
Now, the rings around the eyes....I could be wrong, but I think this is normal as well. I've had bettas who have either had this or developed this before, and it appears to just be something that happens sometimes. I can't say for sure what it is....could very well be a little excess slime coat....but its never bothered any of my guys or led to the development of popeye in my experience. Its just kind of....there.
Hopefully I could offer up something helpful! Good luck!