It's a difficult question of a HUF to answer in a post of a forum. But let me try a bit..........
Answer for 1 : Since the wife is the only solo survivor in that HUF, so the question of a Kartaship doesn't arise in this situation. She will be the solo successor as well as she should bear the liabilities of the whole property.
Answer for 2 and 3 : Even though a Karta can only be the seniormost MALE member of the HUF, The Nagpur High court held the view that mother, through not a coparcener, CAN BE, in the absence of a adult male members, Karta on the Joint family, and her acts will be binding on others as that of a Karta.
In Gangoli v/s H.K. Channappa (See Derrett: Critique of Modern Hindu Law, 117-21), the Karnataka High Court expressed the view that the mother as natural guardian of her minor sons can manage the joint family property and appointment of a guardian by court will not be justified. This is obviously the situation where the father is dead and there is no adult male member.
The SUPREME COURT in Commr. of Income-tax v/s Seth Govind Ram (1966,S.C.24) after reviewing the authorities, took the view that the mother or any other female could not be the Karta of a HUF and therefore cannot alienate the HUF property. This is accordance with the texts of hindu Law. According to Hindu sages only a Coparcener can be a Karta : Since female cannot be a coparcener, they cannot be the Karta of a HUF.
A Junior Male member can also be a Karta in absence of any Senior male member.
So for your Queary relating to that ex client having trading and demat account with you, please note that, A HUF may also contain more then 1 Karta. So for any legal action by you regarding against that HUF, i think, it would be better to target toward both the possible Kartas of that HUF, let the court decide who is who. The Unmarried daughter can be skipped here.