There are actually three methods, which are mistakenly confounded into two by some. The old astrological calculation methods are speculative at best. Multiple people’s naked eye sightings and modern definitive scientific methodologies are definitive.

Either of the definitive methods suffice, though they mostly coincide together. When the new moon is born, the month of Ramadan starts, regardless of you, and based on that birth, it will end, too. Those unable to confirm/see are excused until they do. The same goes when Shawwal enters. Yet, declaring Ramadan is not the function of the individual or the masses, despite the popular hype in Western Muslim communities. It is usually governments and community leadership; unless you yourself witness the month, then you must fast.

The rule is: Fast when people fast, and break your fast (Eid) when people celebrate Eid.

Tirmidhi (Hadith number 802) narrates:عَنْ عَائِشَةَ، قَالَتْ: قَالَ رَسُولُ اللَّهِ ﷺ”“الْفِطْرُ يَوْمَ يُفْطِرُ النَّاسُ، وَالْأَضْحَى يَوْمَ يُضَحِّي النَّاسُ The Prophet ﷺ said: Breaking fast (Eid) is when people do Eid, and Adha is when people do Adha.

Another Tirmidhi narration (Hadith number 697):عَنْ أَبِي هُرَيْرَةَ، أَنّ النَّبِيَّ ﷺ قَالَ” “الصَّوْمُ يَوْمَ تَصُومُونَ وَالْفِطْرُ يَوْمَ تُفْطِرُونَ وَالْأَضْحَى يَوْمَ تُضَحُّونَ The Prophet ﷺ said: Fasting when you all fast and breaking the fast is when you all break the fast, and Adha is when you all do Adha.

The summary: fast when Ramadan is observed by the vast majority in your community, and if you are proven to have missed a day, simply make up the day, and you should be just fine, insha’Allah.

The issue here is what marks [or indicates] the new lunar month. The actual beginning of the lunar month marked by the actual birth of the new moon, or naked-eye sighting that the new moon has been born. Therefore, the method of confirming new moon’s birth may not necessarily be the actual new moon’s birth, as confirmation is capacity-based and method/opinion-based. Obviously, there is only one actual new moon birth. Those unable to align themselves with witnessing the actual new moon’s birth marking the entry and exit of the lunar month, are excused until they do, thus allowing different confirmation of the new moon’s birth, rather than the actual moon’s birth itself. Refer to the previous answer.

The position more aligned with the Prophetic Sunnah is that you pray Tarawih with your family at home. Praying Tarāwīh at the Masjid behind one Imam in the congregation is for those who are weak, demotivated to offer it at home, can’t read on their own, etc. I used to pray it at the Masjid until the Imam finishes, then offer the better prayer at the night of Ramadan which is Qiyām al-Layl reading a full Juz’ (One Qur’anic chapter) at home. Now that I am weak, I pray it at the Masjid in the congregation behind the Imam. Teach yourself and others not to be weak. So, in the West, you can either pray it at home with your family, pray it at the Masjid behind one Imam, or do both. Whatever you choose, do perform the Qiyām al-Layl prayers as it is a confirmed Sunnah. So, either before you sleep, or right before Fajr time enters, offer 2, 4, 6, or more units of prayer.

There is no definitive or conclusive evidence for either, so there is no point in futile discussions.
a- According to the Sahih collections of ̣Hadīth: the Prophet ﷺ prayed 8 units then Witr.
b- And according to the Sahih collections of ̣Hadīth: the Prophet ﷺ prayed 11 units, and the narration didn’t mention Witr specifically.
c- And according to the Sahih collections of ̣Hadīth: the Prophet ﷺ prayed 13 units without mentioning the Witr specifically.
d- And according to Musnad Ahmad: the Prophet ﷺ prayed 16 units without the Witr (yet the Sanad is lightly weak). So for this optional prayer, pray whichever you choose, from 8 to 40, for Allah Ta’ala will not punish you for praying. Those who consider this Bid’a, take heed from what Allah Ta’ala said: أَرَءَیۡتَ ٱلَّذِی یَنۡهَىٰ﴾ [العلق ٩]﴿عَبۡدًا إِذَا صَلَّىٰۤ﴾ [العلق ١٠]﴿ Which means: “Have you seen the one who forbids a worshipper from praying?”

Scrupulousness should lead you to stay from that or from eating once the Adhān started. But if someone needs to take their medications and the Adhān is being called, then they should do that, and that is fine insha’Allah.

There are authentic narrations indicating hastening in breaking the fast, and there is the Qur’anic verse stating: ثُمَّ أَتِمُّوا۟ ٱلصِّیَامَ إِلَى ٱلَّیۡلِۚ ﴾ [البقرة ١٨٧]﴿ (Continue fasting until the night). But when does the night come in? All [scholars] agree that the beginning of the night is when the Sun’s circle disappears from the horizon as a minimum. Therefore, that should be your indicator, and that should be the standard for all calendars. For the longest time I used to pray Maghrib first then break my fast. In times of weakness, I wait for the Adhān to be finished then I break it before praying.

Scholars do not disagree that intimacy with a spouse during the day of Ramadan, except for intercourse, is not prohibited, for that is the only prohibited part. They disagree on whether or not discharge from the reproductive system invalidates fasting. If we overweigh the non-conclusive evidence to our understanding, then a deliberate discharge or acts leading to it may invalidate the fast, whether that happened with a spouse or without.

There is no specific authentic and conclusive evidence singling out one night versus another, except the known Hadith narrations that say it is in the last 10 nights, the odd nights of the last ten nights, etc. To me, Laylat ul-Qadr varies in occurrence after the first one ever in which the Qur’an was revealed, and may not be on the same night every year.

You may give your Sadaqat Al-Fitr in either way, depending on the needs of the receiving needy Muslim.

Scholars disagree on the Wujub or Sunniyyat of Eid Prayers. As for the Khutba of Eid, it is Sunnah. For those who sit and listen, it’s a blessing. Those who leave after the prayer are okay, and no accountability is sanctioned. If the Imam performs a 2-part Khutba (like Jumu'ah) it’s a great blessing. However, it is permissible if he only offers one Khutba.