Sunnah & Nafl Fasts

Fasting on Mondays

Q: Is there any truth to the claims that Nabi (sallallahu alaihi wasallam) fasted on Monday because he was born on a Monday. If it is true, is it permissible to use this to say that one can celebrate his birthday?

Fasting on only the 10th of Muharram due to some valid excuse

Q: I have read that it is makrooh tanzihi for one to fast on only the 10th of Muharram (without also fasting on the 9th or 11th). My questions are:

1. What is the ruling if one could only fast on the 10th due to some valid reason?

2. Should he rather not fast on the 10th?

3. What constitutes a valid reason?

4. What's the meaning and ruling of makrooh tanzihi?