What are b-day cards used for and how do I use them

They were used for the Birthday event. If you collected over 1000 then you got a maximum of 200 gems (there were prizes for other levels too). You can sell the tickets now because the event is over.

@Steven-Cross
On Past Monday 10 July didn't you get a pop up with gem reclaim button ?

Note sure how long the tickets will be valid , i will ask and you can send a ticket to sp about it, this is how
https://forums.socialpointgames.com/topic/30/how-to-report-game-problems-to-support