Whisper Odinsky said:
To try to answer your questions, best I can:
1. Yes, the betta does think he and the gouramis are the same species, which is why they will regard each other as competition and even kill each other for territory. Even if they don't directly kill each other, the stress of being in each others company could well cause them to sicken and die.
Gouramis and bettas are all highly territorial and apart from a mated pairs, you need a big tank with plenty of hiding spaces to keep more than one. If you want to keep one gourami, or a m-f pair, they'd be ideal tankmates for the tetras. I don't know about gouramis with angelfish - I've never known anyone keep them together so you'd have to get advice.
2. Bettas shouldn't be kept with tetras as a rule because the tetras regard his fins as some kind of lure, and will bite him to shreds. Today I bought a "rescued" betta who'd been kept in a tank with tetras. He's a little raggedy but fortunately the LFS owner moved him into a safer tank in time. The only exception to that that I know of are black neon tetras, who seem rather un-tetra-like in their behaviour. Probably because they're baby piranhas (joke!).
3. Bettas and angelfish are totally incompatible.
4. If you could identify the species of catfish you've got, that would help a lot. If it's a big pleco you could develop problems later on as he grows. As a species, most plecos are fine with angelfish, gouramis, bettas or tetras.
5. IMHO, you should get rid of the CAE ASAP. I'm told they're fine in a tank on their own, with a few dither fish and have great personality. I disagree.
You've basically got the contents of several tanks there. Here are some ideas, but they rather depend on the size of your tank (remember the 1 inch per gallon rule - except that angels need more and deeper):
2 angelfish
danios
head-and-tailight tetras
serpae tetras
small pleco or pair of dwarf plecos
?gouramis (check out compatibility with angels - try the cichlid forum).
To that you could add a school of corys if you have room, but don't add any more dither fish. If you want more fish, increase numbers in the schools you already have, rather than buying more species.
The betta could be housed either on his own in a small tank, or in a 10 gal with some danios, rasboras, corys, otos, or cherry barbs (but definitely not tiger barbs), or anything small and gentle.
P.S. I'm not surprised your "cats" never come out - I think the only thing that could make it worse for them is if you bought a red tailed shark and a school of tiger barbs!