First of all, great job answering all my questions in one go! It may sound silly but that really helps move things along instead of going back and forth. Since you seem to really want to learn how to better take care of your new Betta I might give you more info than you want. lol
Once you get your test kit you will be able to monitor the water conditions and that will tell you how much and how often you need to change water (by monitoring nitrates) but 30% once a week is a really good start. But right now, especially because it wasn't cycled and he is sick I would suggest daily WCs. You have provided a much better home for him *claps*. Do you use a dechlorinator/conditioner additive? Which kind? Something like Prime will work as a dechlor and neutralize ammonia, nitrite and nitrates while the tank is cycling. Do you suck out the "gunk" from the bottom when you do WCs?
Sorry to tell you this but that food is horrible. Bettas are so prone to digestive issues and disease that a good quality food is essential. If you look at the ingredients you will see x meal, x gluten, x flour etc which are OK at the very end of the list (least included) but not at the top (most). Things like fish or shrimp meal are the very bottom of the barrel scraps too, look for krill meal or herring meal, something with a name before it. As a everyday staple I'd recommend New Life Spectrum Betta. It is more expensive but it is WAY better for him, the smallest size will last a very long time and he will have less health issues saving you money (and stress) in the long run. 3 pellets is a bit light on feeding but not bad for now with an uncycled tank but long term aim for like 5-10 pellets a day, preferably broken in 2 or more meals. As a treat you can give him things like bloodworms (dried, frozen etc.) and peas. Some people also like to fast Bettas 1 day a week. As you get to know the Betta you can determine if you want to do this, nothing wrong with it.
Any adjustments to lower flow? There are ways to divert the flow for bettas but I can't tell what kind of output it is to give suggestions.
If it truly is fin rot then the method described above is a good coarse of action but rinsing everything in hot water will reset your cycle completely as it will kill all the beneficial bacteria. Just keep that in mind is all. Diagnosing disease is incredibly hard, particularly over the internet, so most of us experienced hobbyist will be VERY hesitant to straight out say "this is what it is, do this to fix it". Best we can do is give you as much info as possible and in the end you decide the proper course of action. Have you done a WC or 2 since this thread started? Any changes in his appearance or behavior?
