Here is a partial answer.
Upto how many days does it extend ?
The number of days for which the impurity is extended remains the same for both Mritaka (death) and Sutaka (birth).
After a death, a ten-day period of impurity is enjoined on those
belonging to the same ancestry (the verse is repeated in Apasthambha).
17 Relatives up to the sixth degree, it is said, fall under the
category of common ancestry, 18and in the case of married women, up to
the second degree. Others should perform the funerary rites of
married women; and such women should perform them for those people.
20 The very same rule regarding impurity applies also to a birth for
those who desire perfect purity.
21 alternatively, it applies only to the mother and the father, or,
according to some, only to the mother, because the birth results from
23 Now, they also quote: At a birth there is no period of impurity for
the man, unless he comes into contact with her.
A Brahmin becomes pure after ten days, a Ksatriya after fifteen days,
a Vais´ya after twenty days, and a Sudra after a month.
Vashistha Dharma Sutras 4.17-38
What are prohibitions on it?
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.
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.
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
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
Datvaa pradakshi - nam kuryaat Suryam dhyaayet
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.