My husband is in the UK on a two year spouse visa - what you refer to as a probationary visa.
My husband and I are quite lucky in the fact that we have our own flat, I'm sure it is very difficult for your friend and his wife living with relatives at the beginning of their marriage. Marriage can also be a bit of a shock to the system in any situation but when one partner has moved to another country it must be really difficult to adjust initially. I think after only two months they really need to try to spend some time talking about what they're not happy with and taking steps to make changes so they can give their marriage a chance.
Your friend's wife may just need a few days on her own to think things through. A short separation should not impact an eventual ILR application at the end of the two year spouse visa, as long as the couple can prove they were in a relationship and not claiming public funds.
I have never come across any Home Office information relating to what each parties rights are if a separation occurs before the end of the two year period, though you might try searching their website
http://www.ind.homeoffice.gov.uk.
She then stormed off to the housing department and claimed homeless assistance.
As far as I am aware every spouse visa is issued with 'no recourse to public funds'. This means the non-uk spouse is not entitled to apply for any public funds including housing, income support, etc. Your friend as a UK citizen is entitled to apply for public funds for himself but not additional requirements for his spouse/spouse's child. This is why documentation is needed to prove accommodation and financial situation before a spouse visa is granted. If you can't prove you can support yourselves your application will be rejected.
I would imagine the housing department need proof of eligibility to apply before granting any public funds to your friend's wife, I don't know what their provisions are for emergency situations when someone is temporarily homeless with a child.
Sorry I couldn't answer your questions, perhaps one of the moderators have experience in this area?