Salatul Layl

Salatul Layl was practiced by the Prophet (SAWA) and Imam Ali (AS) and Lady Khadija from the beginning of Islam. Sincere believers used to do it as a recommended worship. We read in Quran (Verily, your Lord knows that you stand in praying at night, less than two thirds of the night or half of the night or a third of the night and also a number of believers with you) Sura Al-Muzzammil, verse 20. Sura Al-Muzzammil was from the very early Suras revealed on the Prophet (SAWA) in Makkah.

Wassalam.

We can pray for non Muslim parents, relatives and friends who are still alive, to Allah (SWT) to guide them to The Right Path and enlighten their hearts and minds with the light of the Truth.

For those who passed away as non Muslims, we can pray to Allah (SWT) to deal with them with His Mercy.

We are not allowed to seek forgiveness for those who insisted and died on Shirk (polytheism) as we read in Quran, Sura Tawbah, verse 113: It is not proper for the Prophet and the believers to seek forgiveness for the Mushriks, even though they be of kin.

Wassalam.

Yes. Recommended Prayers especially Salat al-Layl ( Night prayer) is highly recommended for every one including those who have unfulfilled Qadha  prayers on them. No doubt, the reward of performing Qadha prayers is much more than the reward of recommended prayers. 
You must perform every prayer within its time and never miss any obligatory prayer.

Wassalam.

Yes we can repeat recommended prayer but one time is for us and other times with the intention of on behalf of others like parents, relatives and friends whether alive or dead.

Wassalam.

Tahajjud time is during the night ( starting from either midnight or even after Isha) and ending with Fajr time. Tahajjud can be also performed after Fajr as Qadha.

Wassalam.

Tahajjud is a great recommended worship for every night of the whole year. It is also called Salat Al-Layl or Namaz Shab. It is performed individually and not with congregation. Umar Ibn al-Khattab ordered during his government to pray Tahajjud with congregation which was then called Taraweeh.

Taraweeh is a Bid'ah which was never practiced by the Prophet Muhammad (SAWA) but was initiated by Omar Ibn al-Khattab in year 14 after Hijra.

Wassalam.

http://www.duas.org/tahajjud.htm

Namaz Tahajjud can be performed either fully as 11 Rak'ats or shortly as three Rak'ats (2 Shaf' and one Witr) or in the shortest way which is just one Rak'at which is called Witr.

Wassalam.

Yes, we can pray for our Sunni friends to be granted the light of following Ahlul Bayt (AS) who are the real source of the authentic Sunnah of the Prophet Muhammad (SAWA).

Wassalam.