Currently there are 65 different species. The War Hammerhead Shark is currently unavailable, probably until Shark Week next year. The Gobblerfish from what I understand will be gone at the end of this month. The other 63 species should be available year round.
To display, there really isn't a best, it's about what you really like and want to have on display.
The rare or epic fish are the ones that bring in the most gold, but to be honest I have not paid attention to how much I'm selling them for. I just keep clicking check boxes until I've selected all the fish I want to sell and then sell them... unless I have too much gold and then I just start throwing them away.
The rarest ones I'm finding so far are the Jolted Dekoi and Errol Flynn. I've got 12 wizards who have all caught at least one of 61 different species. Of the remaining 4 species, 10 have caught the Krokotopian Eel, 8 have caught the Cranky Catfish, 1 has caught an Errol Flynn, and none have caught a Jolted Dekoi.