What does 1) A la tercera va la vencida
2) Ya me doy cuenta
3) ¿A que no?
4) ¿A que sí?
5) ¿Adivina?
6) ¡Anda! mean?

See a translation

1. Third time's a charm. Two efforts at something have failed but perhaps the third chance will be successful.
"He suspendido el examen de conducir, pero creo que a la tercera va la vencida".
2. When you say that means that you suddenly realize or notice something. It is like saying "I see".
"No tenía claro sus intenciones, pero ya me doy cuenta"
3. It is like challenging to someone.
"¿A qué no te atreves a llamar?"
4. It could be the answer of (3) or it could be an expression to request someone's corroboration.
"Es buena idea, ¿a qué si?"
"It is a good idea, isn't it?"
5. Guess what.
"¿A qué no sabes quién viene hoy? Adivina."
6. Express some kind of surprise.
-"Viene mi hermana.
-¡Anda!"