We advise you to depend on Almighty Allah in all your acts and enter the exam room with the confidence that He is with you as long as you are acting upon His orders and completing your duty of fasting. Yes, if you can travel on the exam day after dawn and break your fast after reaching the Sharia-prescribed distance of breaking the fast, then you can break your fast that day as a traveler, and you should then make it up afterwards.
Engaging in such activities on the night or day of Ashura is not in accordance to what has been narrated from the Imams (peace be upon them) in the exhortation to express grief and sadness over the tragedy suffered by Imam Hussain (peace be upon him), his household and companions on the day of Ashura.
The said prayer is not obligatory, and it can be repeated as many times as one desires. An individual is expected to feel the Godly bounties that surround him day and night. Such feeling is not limited to the time after granting the wishes. It was reported that it is recommended for an individual to say everyday seven times "Alhamdulillah ala kulli ne'matin kanat aw hiya kaainah" (Praise Allah for every bounty that has been granted and for those which will be granted)".
The popularity of the concept of taqlÄ«d amongst the masses in these days is due to the broad existence of means of communication which have led to the wide spread of knowledge of religious laws. It is not because the concept is new. Rather, it was present at the time of the Imams (peace be upon them) when they were directing their companions and followers, in general, to learn the religious laws. This naturally led to many people asking questions about their affairs, and it was not practical for all of them to reach the Imam (peace be upon him) and ask him in person, so they used to refer to his close companions who could be reached to gain knowledge in religious laws. There were known narrators who were giving edicts to people and the Shias used to refer to them during those days, like Muhammad bin Muslim, Zurarah, Zakariyya bin Adam and others.
This need continued to exist after the occultation of the Twelfth Imam (peace be upon him) until now.
It is a natural need and a sensible process, as people refer to the experts in various fields like medicine, engineering and other kinds of expertise, as people are not satisfied with their own limited knowledge in these areas.