Wa `alaykum As-Salamu wa Rahmatullahi wa Barakatuh.
In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
Dear sister in Islam, thanks a lot for your question, which reflects your care to have a clear view of the teachings of Islam. Allah commands Muslims to refer to people of knowledge to become well acquainted with the teachings of Islam in all aspects of life.
As for your question, spouses should solve their problems through mutual consultation, respect, kindness, and love. If they fail to solve their problems themselves, they can resort to arbitration or professional Islamic counseling. Spouses may perform the Prayer of Need or Salatul-Hajah so that their problems would be solved by the help of Allah.
Focusing more on your question, Sheikh Ahmad Kutty, a senior lecturer and Islamic scholar at the Islamic Institute of Toronto, Ontario, Canada, states:
The proper thing for spouses to do when marriage is not working out is to seek marital counseling from those who are properly qualified in the field. If that does not work, then they should resort to due process of arbitration, involving wise and knowledgeable people from both sides. However, it is equally important to turn to Allah for guidance through Prayer (salah) and du`aa’ (supplication), as we must do in all things. We read in the sources that the Companions of the Prophet (peace and blessings be upon him) would turn to Allah for succor even in the most trivial matters of everyday life.
Having said this, however, I must add that Istikharah must be resorted to when one is making a final decision whether to remain married or part company; otherwise, one should offer Prayer of Need (known as Salatul-Hajah).
As for your second question, there is no precedent in the sources that Istikharah was made by people on behalf of others. But still one cannot say it cannot be done; since there is nothing prohibiting it, we are not allowed to prohibit it. Since Istikharah is essentially a prayer, there is precedent for offering prayer on behalf of others, so it should be fine to do so.
Having said this, however, I must still add that the proper and ideal thing to do is for the person or persons involved to perform Istikharah themselves instead of depending on others to do it on their behalf. That is the Sunnah way in such matters. Once they have done it, in addition, they can request other people or parents to perform it as well.
Allah Almighty knows best.