flr(O) mother of a british citizen child

[samixade]

Hello good people of this forum. I need urgent help for a dear one.
She has a baby for a british citizen and which automatically qualifies the baby too as a british citizen.
The mother came into UK in 2006 on a visiting visa which as expired for a long time now and she as already overstayed for 6+ years.
she was in a relationship before she found out that the man in question was already married and that the relationship won't be going anywhere and she broke off the relationship when the whole truth came out. Though, the father's name is on the birth certificate and the baby holds a british passport but she actually cares for the baby alone and she doesn't receive any support from the father and she only feeds the baby with money she has saved up over the years whilst working illegally and definitely with a different name.
MY QUESTIONS ARE:
(1) can she apply based on appendix fm?
(2) what documents are required from her?
(3) presently squatting with a friend but can she get housing benefit to support her till she can work again (mind you, she hasn't got NI number linked to her real name)

Lastly, please no insult as some might suggest she got pregnant because of right to stay in UK.

I await your noble responses as they will help guide towards the right direction.

[Amber]

Consider FLR(O) - family life leave:

Consider:

EX.1 This paragraph applies if
(a) (i) the applicant has a genuine and subsisting parental relationship with
a child who-
(aa) is under the age of 18 years;
(bb) is in the UK;
(cc) is a British Citizen or has lived in the UK continuously for
at least the 7 years immediately preceding the date of
application; and
(ii) it would not be reasonable to expect the child to leave the UK;

See the Guidance (click) for further details.

[samixade]

Please, what if the mother came into UK with visiting visa?
secondly, does she need to sit English language test as well?
Thirdly, how long does the application take and what documents are expected to be submitted with the application?
Please your answers will be so much appreciated.