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.
What is the ruling on reciting aloud (*jahr*) in a silent (*sirriyyah*) prayer?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah.
Reciting aloud (jahr) in the Fajr prayer, in the first two rakʿahs of Maghrib and ʿIshāʾ, and reciting silently (isrār) in the Dhuhr and ʿAṣr prayers is a sunnah. If the worshipper recites aloud where silence is prescribed, or the reverse, his prayer is valid, although it is better to follow the sunnah, in imitation of the Prophet (peace and blessings be upon him). Abū Hurayrah (may Allah be pleased with him) said: “In every prayer there is recitation. What the Messenger of Allah (peace and blessings be upon him) let us hear, we let you hear, and what he concealed from us, we conceal from you. If you add nothing to Umm al-Qurʾān (al-Fātiḥah), it suffices, and if you add to it, it is better.” Narrated by al-Bukhārī.
Shaykh al-Islām Imam al-Nawawī (may Allah have mercy on him) said in al-Majmūʿ (Vol.3/389): “The sunnah is to recite aloud in the two rakʿahs of Ṣubḥ (Fajr), Maghrib, and ʿIshāʾ, and in the Friday prayer, and to recite silently in Dhuhr, ʿAṣr, the third rakʿah of Maghrib, and the third and fourth of ʿIshāʾ. All of this is by the consensus of the Muslims, together with the mutually reinforcing authentic ḥadīths on the matter. This is the ruling for the imam. As for the individual worshipper (munfarid), it is sunnah for him to recite aloud according to us and according to the majority… If he recites aloud where silence is prescribed, or the reverse, his prayer is not invalidated, nor is there a prostration of forgetfulness (sujūd al-sahw) for it, but he has committed something disliked (makrūh).” And Allah Almighty knows best.
What is the ruling on seeking forgiveness between the two Friday sermons?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
It is permissible to seek forgiveness (Istighfar) between the two Friday sermons (Khutbatayn), and it does not invalidate the Friday prayer (Salat al-Jumu'ah). Furthermore, it is not disliked (Makruh) to speak before the sermon, after it, or between the two sermons. Likewise, it is not disliked for one who enters during the sermon to speak if there is a need for it, provided he has not yet taken a place and settled into it. And Allah the Exalted knows best.
Is "Bismillāhir-Raḥmānir-Raḥīm" a verse (āyah) of Sūrat al-Fātiḥah?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah ﷺ.
The basmalah ("Bismillāhir-Raḥmānir-Raḥīm") is, without question, a verse (āyah) of the Noble Qur'an, and according to the Shāfi‘ī school, it constitutes a verse of Sūrat al-Fātiḥah itself. Al-Fātiḥah is a pillar of the prayer, and it consists of seven verses, of which the basmalah is one. The prayer is not valid without reciting it, nor if any of the Fātiḥah's verses, letters, or shaddahs (consonant doublings) are omitted or altered.
It is stated in Bushrā al-Karīm (Vol.1/p.202), from the books of the Shāfi‘ī school: "The basmalah is a complete verse of it [i.e., of al-Fātiḥah] in practice — and it suffices, for this to be established, that there be strong probability (ẓann), especially given that this has come very close to certainty (yaqīn), due to the consensus (ijmā‘) of the Companions on its inclusion in the muṣ-ḥaf [the written text of the Qur'an] in its script, despite their extreme diligence in excluding from it anything that did not genuinely belong to it — to the point that they even [carefully regulated] its diacritical dots and vowel markings." And Allah, the Exalted, knows best.