Assuming you are entitled to legal aid, as I've said (29 Jan 01.36) you are best advised to go for the jugular divorce now on the grounds of her unreasonable behaviour, desertion, and adultery before the rules on entitlement change.
With regard to the rent arrears, I would have preferred to see a different outcome whereby the HA pursue your w as the former sole tenant of the property for any sums outstanding.
There should be no difficulty in respect of the HA transferring the tenancy into your sole name immediately as, were they to succeed in repossession proceedings, your local authority will be statutorily obliged to house you and your sons and in such cases commonsense usually prevails.
With regard to your w's seeming attempt to cherrypick which of your sons she sees at any given time, I would suggest you make it clear to her that this is a case of all for one and one for all and any contact which takes place in or outside of your home will not be subject to her, whim but will be conducted in the best interests of the dc.
You're still reeling from the shock and the extent of your w's duplicity and there's not a lot anyone can say or do to speed you through the healing process - but it will happen and, in the meantime, I would suggest you confine any conversations you have with her to matters relating purely to contact with the dc and their general wefare and wellbeing.
A solicitor can advise what you may expect in terms of division of any jointly owned property and child support, but I suspect you may need to file a claim with the CSA for cs payments which can be taken at source from your w's wages/salary - this may give you some scant consolation if you should see her working.
I'm sorry to say that you should prepare yourself for the possibility that your w may have another child with the om sooner rather than later and may subsequently claim that she is best placed to be the primary carer of your dc. This possibility is all the more reason why you should let your head rule your heart and act now, both in respect of divorce and in estabishing residency of the dc.