So the other day I accidentally used the Cyclone Shout on my carriage driver who's always in front of my house in Falkreath. This resulted in my housecarl Lydia to attack me. I jumped on the carriage and let my follower Teldryn Sero handle this. This resulted in quite a big fight between Lydia and Teldryn Sero. In the end Teldryn won the fight by stabbing an ebony sword through my housecarl Lydia. Naturally she died. No big deal really, but i need a new housecarl for my house in Falkreath. How can I do that?

Take another follower and see if they can become a steward.

I've tried that in Morthal with whats his name? Valdemar. I did not get the dialogue to make him my housecarl in Falkreath. Maybe the Morthal housecarl only works when you build a house in Morthal. This is very annoying!

EDITED:

Should I try and send my current follower away and try it again?

You probably mean steward - the housecarl in Falkreath is Rayya, and you can never assign a new housecarl to a house. The name housecarl is actually misleading, I think that's why so many are confused about this - the housecarl is more of your personal bodyguard than someone who comes with the house. You get the housecarl assigned to you by the Jarl when you become Thane - but in most holds (Whiterun being the exception) you must purchase a house to be able to become Thane.

The steward is the one YOU appoint to take care of your house. Many, but not all, followers can be stewards, so try another one.

i would just try another, this happened to me before. i noticed that they dont always ask about becoming steward the first time you take them there. some times after quite a few visits