What is Tarawih prayer?
Tarawih prayer is the night prayer (Qiyam) performed during the month of Ramadan. It is called "Tarawih" because Muslims used to prolong the standing (qiyam) in it and take brief rests after every four rak‘ahs.
What is the ruling of Islamic Law on supplicating with other than the transmitted (Ma'thur) supplications in prayer?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah.
It is permissible to supplicate (make du'a) during the prayer for anything from the affairs of religion or worldly life, and the prayer is not invalidated by doing so.
Shaykh al-Islam, the Imam an-Nawawi, may Allah have mercy on him, says: "Our madhhab [i.e., the Shafi'i madhhab] holds that it is permissible for a person to supplicate during it [the prayer] with anything that is permissible to supplicate with outside the prayer, from the affairs of religion or worldly life. He may say: 'O Allah, grant me wholesome sustenance, offspring, a house, and a beautiful young wife' — describing her; or: 'O Allah, free so-and-so from prison,' or 'destroy so-and-so,' and other such things — and none of this invalidates his prayer according to us. And this is also the view of Malik, ath-Thawri, Abu Thawr, and Ishaq." [Al-Majmu', Vol.3/P.454]. And Allah, the Exalted, knows best.
Should a person feel pleased, or have a virtuous vision after offering Istikhara (guidance prayer) in order to do what he/she had offered it for?
The result of the Istikhaarah is not necessarily that a person sees something (in his dream), or feels pleased, but the most important result of the Istikhaarah is whether a person is enabled to do a given matter or not.
How can I love the prayer (Salah) and start praying?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
Love for the prayer (Salah) comes when you realize that this world is fleeting and the Hereafter is everlasting. Your prayer is the righteous deed that will comfort you in your grave and be your salvation before Allah in the Hereafter. Do you not wish to attain tranquility in this life and Paradise in the next? Reflect upon your state and do not delay, for life does not wait. We ask Allah to grant us and you steadfastness (Thabat).