All tetras, barbs and Corydoras need to be in groups consisting of at least 6 (preferably 10) or more individuals. Get 6 embers, and maybe 4-6 neons. The tank size limits you to the number of fish and probably isn't big enough for more Cories or cherry barbs. And if the cherry barbs are happy (displaying and not fighting) they are probably ok in a group of 5.
Check any new fish at the shop before you get them. Look for pale colours; cream, white or grey patches; red patches; white lips/ mouths; unusual spots, etc. If any fish look odd, don't get any fish from that tank. the last thing you want to do is introduce a disease into an established aquarium.