1) Is there any problem praying in the Ommeyad’s mosque in Damascus while travelling there? Is salat valid in it?

2) If I go over there just as a touristic visit, do you think I should nonetheless pray the two rakats of entering a mosque, or it is better not to do so?

1) It is best not to with the shrine of Sayyida Zaynab (p) present there

2) It is only recommended to do that, so you may do so or not