Now, how many commentaries are there of these Sutras? Are any of these commentaries translated in English and available free online?
I know of six commentaries on Gautama's Nyaya Sutras:
- Vatsyayana's Nyaya Sutra Bhashya, which you can read here.
- Uddyotakara's Vartika on Vatsyayana's Bhashya, which you can read in four volumes here, here, here, and here.
- Vachaspati Mishra's Tatparya Tika on Uddyotaraka's Vartika, which isn't available in English, but is available in Sanskrit here.
- Udayana's Parishuddhi on Vachaspati Mishra's Tatparya Tika, which isn't available in English, but is available in Sanskrit here.
- Raghunatha Shiromani's Bhashya Chandra, which isn't available in English, but is available in Sanskrit here.
- Vishwantha's Nyaya Sutra Vritti, which you can read here.
One note of caution: not all these commentators are actual members of the Nyaya school. Vatsyayana and Uddyotakara were Naiyayikas, but Vachaspati Mishra was an Advaitin, and Udayana and Raghunatha Shiromani were members of the Navya Nyaya school, which was a fusion of Nyaya and Vaisheshika. The original Nyaya school died out around the 12th century, so all commentators after that were from outside the Nyaya school.
I know about two English translations with commentaries which are freely available online. I think you will get it in archive.org
The aphorisms of the Nyaya Philosophy by Goutama with illustrative extracts from the commentary by Vishwantha (in Sanskrit and English), Allahabad, 1850.
The sacred books of the Hindus Vol VIII: The Nyaya sutras of Gotama. Translated by Mahamahopadhyay Satishchandra Vidyabhushana, Principal Sanskrit College, Kolkata, Revised and Edited by Nandalal Sinha.
There have been a few commentaries on the Gautama Nyaya Sutras. The names are given in The Nyaya sutras of Gotama. Translated by Mahamahopadhyay Satishchandra Vidyabhushana.
The book lists 15 commentaries on the Nyaya Sutras.
- Nyaya Bhashya by Vatsayana otherwise known as Pakshila swami.
- Nyaya Vartika by Udyotakara
- Nyaya Vartika Tatparya Tika by Vachaspati Mishra
- Nyaya Vartika Tatparya Tika parishuddi by Udayana
- Parishuddhi Prakasha by Vardhamana
- Vardhamanendu by Padmanabha Mishra
- Nyayalankara by Srikantha
- Nyayalankara Vritti by Jayanta
- Nyaya-manjari by Jayanta Bhatta is available in Sanskrit on archive.org
- Nyayavritti by Abhayatilakopadhyaya
- Nyayavritti by Vishvanatha
- Mitabhashini Vritti by Mahadeva Vedanti
- Nyaya Prakasha by Keshava Mishra
- Nyayabodhini by Govardhana
- Nyaya Sutra by Mathuranatha.
This pdf by Shodhganga (from page 22) gives many more sub commentaries and independent works besides the ones stated above.