Conditional divorce in a letter

Q: Some months ago my wife got angry due to some family issues and went to her parents’ home but did not come back. I tried to bring her back but she didn't come. One day I sent her a letter that If you want to come then welcome a thousand times from my side otherwise consider this letter a divorce from my side. (I mentioned Talaaq, Talaaq Talaaq in letter). Decision is in your hands. Now I want to know whether divorce has happened after she got my letter or not?

Thoughts of showing off

Q: If you get thoughts, urges or daydream about showing off your worship to other people so they will praise you but you don't act on the thoughts or urges, would you be still a sinful? Basically they are just thoughts, desires and urges of showing off in your heart but you don't physically show off.

If you get thoughts or daydream or images in your mind of shirk and kufr but never act on them physically, they are just in your mind, would you come out of Islam or become a disbeliever? 

Mazoor making tayammum

Q: I have some health issues and I have to renew wudhu before a salah starts because I am a mazoor or excused person (urine comes out). Can I do tayammun instead of wudhu? Because it takes me 10 minutes to do wudhu and I am not well plus if I wash my body parts too much they get dry.