Is it permissible for a person, who vowed to offer two Rak`ahs (units of prayer) everyday for Allah`s sake, to do so after the Fajr (dawn) prayer?
In principle, a vow should be fulfilled as it was originally intended, and if its time lapses then, it should be fulfilled at any time because of the vow. Therefore, it is permissible to offer them after the Fajr prayer, and there is no prohibition in doing so. However, it is preferable for him to offer these two Rak`ahs at some other time to avoid the disagreement amongst the scholars in this regard.
What is the ruling on having brotherly ties between a strange man and a strange woman, and is the former considered a Mahram (unmarriageable) to the latter?
Such a relation between a strange man and a strange woman is forbidden in Islam because it involves forbidding what Allah has allowed by marriage, and allowing what Allah has forbidden such as looking and the like. Such a relation doesn`t render any act lawful between them, thus they are forbidden to look at each other, have a Khulwah (seclusion), and travel together.
Is it permissible to repeat the Istikhara (seeking guidance) prayer, and is it a condition that one should sleep after offering it?
All perfect praise be to Allah,The Lord of The Worlds Istikhara is offered once, but it is permissible to repeat it, and sleeping after offering it is not a condition for its validity. And Allah Knows Best.
Is there a solution to put an end to thinking of worries during prayer?
Firstly, make ablution after the time for prayer begins and busy yourself with remembering Allah, The Exalted, before the iqaamah (start of prayer).In order to become humbly submissive during prayer, you must perform prayer in congregation. After the prayer ends, stay in your place and read the post-prayer remembrances (Adhkaar) and make supplication to Allah, The Exalted.