What is sutak? 

Generally there are various prohibitions when a child is born into a family. It goes up to several days like 11 days which is called the Sutak period. There are also various community rules regarding no. of days.

Which scriptures mention about Sutak dosha?

Manu Smriti, Chapter 5 declares ten days of Sutaka (impurity) for Sapinda.

1. A man who hears of a (Sapinda) relative's death, or of the birth of a son after the ten days (of impurity have passed), becomes pure by bathing, dressed in his garments.

Vishnu Smriti, Chapter 22 gives impurity period by each Varna.

1. The impurity of a Brâhmana caused by the birth or death of Sapindas lasts ten days.

2. In the case of a Kshatriya (it lasts) twelve days.

3. In the case of a Vaisya (it lasts) fifteen days.

4. In the case of a Sûdra (it lasts) a month.

Gautama Smriti, Chapter 14 agree with Vishnu Smriti. It declares:

1. (The rules regarding impurity caused by the death of a relative apply) to the birth (of a child) also.

And then

1. The Sapindas become impure by the death (of a relative) during ten (days and) nights, except those who officiate as priests, who have performed the Dîkshanîyeshti (or initiatory ceremony of a Srauta sacrifice), and those who are students.

2. (The impurity) of a Kshatriya lasts for eleven (days and) nights,

3. (That) of a Vaisya twelve (days and) nights,

4. (Or), according to some, half a month,

5. (And that) of a Sûdra a whole month

Vashishtha Smriti, Chapter 4 also declares the same.

1. A Brâhmana is freed from impurity (caused by a death or a birth) after ten days,

2. A Kshatriya after fifteen days,

3. A Vaisya after twenty days,

4. A Sûdra after a month.

Parashara Smriti, Chapter 3 allow Brahmana to be pure after three days.

1. Now I shall speak of the term of impurity resulting on the occurrence of a birth, or a death. Brahmans in three days become pure again, when a relative has died, or has been born.

2. A Kshatriya recovers purity in twelve days ; a Vaisya, when fifteen days have passed ; a Shoodra, after the lapse of a month ; this is conformable to what Parasara has said.

What are prohibitions on it?

Vishnu Smriti, Chapter 22 gives following prohibitions:

1. During the period of impurity oblations (to the Visvedevâs), gifts and receiving of alms, and study have to be interrupted.

Most of the Dvijakarmas are to be abandoned except Sandhya-rites which can still be continued with slight variations (like without reciting mantras loudly, without pranayama and reciting sankalpam):

A Vipra, [on the occasion of a birth or death] should refrain, for ten days, from making gifts or studying the Veda's. A Kshatriya is purified in twelve days ; a Vais'ya in fifteen days ; and a S'udra in a month. Such is the deliverance of Samvarta.

A twice-born person should, daily and duly, perform- the five sacrifices. A Vipra, seeking his well-being, should, under no circumstances, abandon them. One should, [however,] always abandon them [in an impurity consequent] on birth and death.

Samvarta Smriti

However one should also note the following verse:

There is no sin, if any intervening impurity, consequent on a birth or death, takes place, in the celebration of a marriage, festivity, or a sacrifice, determined upon before. So Atri has said.

Atri Smriti

As regards proving the fact that Sandhya-Rites can still be performed during these Asuchi days, we can quote from the book Dharma Bindu:

Sutake mritakechaiva Sandhyaakarma samaacharet
Manasocchhaarayet mantraan praanaayaamaadrute |

Sandhyopaasana needs to be performed both during ‘Jaataashoucha’or ‘Mritaashoucha’ or Ashuddhi on account of births or deaths, by maanaso cchaarana or recitation of mantras mentally without resorting to pranaayama, confirms Pulastya Maharshi.

Vyaasa too states: Prakshipet sutaket arghyam Gayatreem samuccharan,
Datvaa pradakshi - nam kuryaat Suryam dhyaayet dvijottaamah,
Dasha krutastu Gayatreem manasaiva Shuchih smaret

During Jaataashoucha too, a Dvija could recite Gayatri and perform by way of Surya dhyaana pradakshinas. During the ten days of Ashuchi, Gayatri should be recited by one’s own mind.

More answers are awaited.