The thing that bothers me the most about Tauriel is honestly her hair color. No, your hair isn’t going to be red unless you’re one of the descendants o Mahtan, who as far as we know only got one daughter: Nerdanel, wife of Fëanor. Meaning that for you to be a red-haired elf and live in…
Actually, red hair was not said to be “only for a certain Noldor family” or anything like that. In all the supplementary materials I’ve read it is mentioned that Nerdanel, Fëanor and a few descendants has red hair, but never that no one else can have red hair (unless they’re pure Vanyarin, which denotes golden hair, but again, we don’t know if strawberry blond, which is a type of red and blond, is counted in “golden”).
Add that the Teleri are the most numerous and have been mentioned having blond (like strawberry blond?), dark (might contain auburn/mahogany which is also the red gene), white, silver, golden silver, graying hair and the gene for beards, them also having some red genes in there isn’t very far-fetched.
TL;DR: Basically, saying that “Fëanor’s family has some red-haired elves” does not mean “Fëanor’s family are the only red-haired elves!”.