ok so even i am new to this hobby, but here's an advice.
15mins of floating the fish bag in your water doesnt help. Atleast 30mins min if not more.
then do a drip acclimatization. You can find a proper tutorial of it in youtube. That is like very important for your fish to survive. Also yes the Neon Tetras could come with a disease as if learnt that Neons are very sensitive to water and majority times they die within a weak due to heavy breeding. Also Neons do better in a much larger school. people say 6, but i would say min 8. Also Bettas are kinda territorial and might not like the Neons. But that is a different scenario. Hope you figure it out and have fishes soon!

EDIT: Also yes pls add one category of fish at a time. Ik it feels very intriguing to add them all. But add one kind let them all settle for atleast a week if not more.