I can tell you he is stunning! I want him. Do you ever call him Alfie? :p
He is a crowntail, but due to the different lighting I can't tell if he is a copper (my favourite - the first photo) or a coppery blue. Which photo is more accurate?
There are a lot of different betta species out there ('wild-type' bettas). LittleBettaFish is the go-to guy on wild-types. :) For our domestic betta spledens, there is a large variety of tail-types.
Veil-tail (VT): most common in shops. The long droopy tail looks like a veil.
Crowntail (CT): what you have. A nice spiky tail. :)
Delta tail (DeT, I think): a wider spread than a veiltail
Super-delta: the tail is wider than a DeT but short of being a halfmoon
Halfmoon (HM): tail is basically a D shape, like a perfect semi-circle
Plakat (PK): short tail, like a wild splendens
Doubletail (DT, I think): Like a delta but the tail is split in half.
There are also a couple of crazy ones like rosetail and half sun. Then you have all the different colours, plus variations like marble, dragonscale, etc!
A normal CT like your boy will generally get about 2 inches in the body, then some extra tail. He could be between 6 months and a year as a petshop betta. You may see some growth, you may see none. I'm sorry I can't be more precise, but it is so dependent on genetics, background, diet etc.
I'm always tempted to call snails Tardis, since there is so much more of them on the inside than you'd think. *nerd alert* I think your tank is nicely stocked. You could have more ghost shrimp, if you like. :)
Just a tip - bamboo is not aquatic and will rot if the leaves are submerged. I also don't think that the plant at the back right of your tank is aquatic, so it may well die too.
If you want to get some real tannins in the water, consider Indian Almond Leaves (IAL). Bettas love them and your shrimp will probably play on them too. :) They harden betta scales and encourage bubblenesting, as well as darkening the water. :) They also make your water changes smell of tea instead of fish poo, which I like.
As for your filter, a little current is fine, as long as he doesn't seem bothered by it. :) My boy Odysseus actually builds his bubblenests around the filter downflow, because he is a silly.
I hope this helps! Lots more pictures of your pretty boy to oggle, please. :)