There are people that swear tanks as small as a gallon can and will be able to hold a cycle but I wouldn't bother with trying. Just do the 2 weekly water changes - one 100% with gravel cleaning and one 50%. Unless you want to go with all live plants. You can also cycle a tank that way but you need certain types of plants to do it right.
I can't tell from the pics but if he does have fin rot - clean, warm water is the best way to go.
I would add some more decor - like a few silk plants and a cave of some sort. I like to have an area with several oversied plants to make it like a forest for them to swim through and one plant that reaches the surface as they like to sleep resting on the leaves near the surface.
Things to keep on hand:
1) AQUARIUM SALT
2) PLAIN UNSCENTED EPSOM SALT
3) Antibacterial Medication - Furan 2 or Triple Sulfa or Kannaplex (most likely to find this one online)
Those 3 things should be enough for almost any ailment your betta will encounter
Oh and welcome to the forum. I am almost postive the majority of us joined because we had no idea what we were doing