How many currently available smritis are there? Are there any smritis whose names are known but not available?
What are the names of smritis available along with the corresponding author names and the times of validity of them?
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There are 18 major Smritis in Hinduism. The Yajnvalkya Smriti chapter 1 verses given below mention the 18 respective authors.
Manu, Atri, Vishnu, Harita, Yajnavalkya, Usana (or Shukracharya), Angira, Yama, Apastambha, Samvarta, Katyana, Vrihaspati, Parashara, Vyasa, Sankha, Likhita, Daksha, Gotama, Satatapa and Vashishta are the promulgators of Dharma Shastras (Smritis).
Times when they are valid:
This is mentioned in Parshara Smriti. For Krita (or Satya Yuga) Manu Smriti is said to be the valid (most effectively applicable) rule book. For Treta Gautama Smriti, for Dwapara Sankha Smriti and Likhita Smriti both and for Kali, the Parashara Smriti is said to be applicable.
Krite tu Manava Dharmastretayam Gowtamah Smritaha ||
Dwapare Sankhalikhitah Kalou Parashara Smritaha ||
In Krita Yuga Manu's laws apply, Gowtama's in Treta, in Dwapara those written by Sankha and Likhita apply and Parashara Smriti is the one that applies in Kali Yuga.
From this page you can download all the Smritis mentioned above plus few others (but in Sanskrit only).
Updating my answer with the names of many other Smritis that are not mentioned in lists given above.
The Padma Purana mentions the 36 Smritis (18 major+18 minor ones) and their authors' names.
The following screenshots are from the "Introduction" of the book "The Dharma Shastra Or The Hindu Law Codes - Chaukambha Amaravati studies (Varanasi)":
So, new names of Smriti-authors are Kashyapa, Pracheta etc but probably these texts are not extant now.
Viswamitra, Devala, Marichi, Pracheta, Narada, etc are the authors of the respective Upasmritis.
And, as far as I know, few of them (like Viswamitra, Narada) are extant even now. But most of them seems to have been completely lost.
Some more authors of Smritis, which appear to be completely lost now, are also found in the same book and are like - Agni, Soma, Prajapati, Buddha, Satyayana.