Taraweeh is the night prayer of Ramadan. The Prophet ﷺ said: 'Whoever stands in prayer during Ramadan out of faith and hope for reward, his past sins are forgiven.'
The Prophet ﷺ prayed eight rak'ahs followed by Witr; twenty rak'ahs became widespread practice from the time of Umar (may Allah be pleased with him).
Both are valid. Choose what allows you to pray with calmness and presence of heart.
A short rest is taken after every four rak'ahs, which is where the name Taraweeh comes from. Use it for dhikr and quiet supplication rather than idle talk.
Finish the night with Witr, and if you intend to pray tahajjud later, delay Witr to the end of the night.
اللَّهُمَّ اهْدِنِي فِيمَنْ هَدَيْتَ وَعَافِنِي فِيمَنْ عَافَيْتَ وَتَوَلَّنِي فِيمَنْ تَوَلَّيْتَ
Allahumma-hdini feeman hadayt, wa 'afini feeman 'afayt, wa tawallani feeman tawallayt
O Allah, guide me among those You have guided, grant me well-being among those You have granted well-being, and take me into Your care among those You have taken into Your care.
Sunan Abu Dawud 1425 — sahih
سُبْحَانَ الْمَلِكِ الْقُدُّوسِ
Subhana al-Malik al-Quddoos
Glory be to the King, the Most Holy.
Sunan al-Nasa'i 1699 — said three times, raising the voice on the third
Yes, it is valid at home, though praying it in congregation behind an imam carries the reward of standing the whole night.
Yes. You may pray extra night prayers after Witr without repeating Witr, though delaying Witr to the very end of the night is better if you can.
It is a virtuous practice, not an obligation. A shorter recitation prayed with concentration is preferable to a rushed complete reading.