Apologies for the length of this, but it's complex!
Capital from a house sale can be disregarded for a year, but only when it is to be used to buy another home.
When you have capital of over £6,000, they deduct £1 pw from housing benefit for every £250 or part thereof from £6,000-£15,999. At £16,000, you get no housing benefit at all. So, if you're left with £15,000 after clearing your debts, they'd deduct £36 pw from your housing benefit (£15-£6k = £9k/250 = 36).
However, as you won't be in work, I'm guessing you'll either be on income support or claiming carer's allowance plus income support (around £34-35). The tariff income at £15k will wipe out your entitlement to income support, but you'll only lose a tiny bit of housing benefit. As your savings dwindle, and you become entitled to income support, no matter how small the amount, you'll get full housing benefit, unless you're benefit capped.
If you don't get DLA for your SN child, your benefits will be capped. If so, your total benefit entitlement, including your housing benefit, will be restricted to £350pw (more if you're in London). Anything above this will be deducted from your housing benefit. There may be an exemption for 39 weeks, eg if your ex has worked for the last year.
Many of the families I support are left with very little to live on after rent, because a 3 bed house in my area is around £250 pw.
Don't go and blow loads of money though. If the council (or the DWP) think you've deliberately spent funds to make yourself eligible for benefit, or increase the amount you get, they can treat you as though still have the money and reduce or withhold benefits entirely.
With regard to the housing register, anyone can be on the register, as long as they meet the residence qualification (this varies, but 2 years residence in the council's area is fairly usual). Whether they will decide they have a statutory duty to help you is another matter.
I live in an area where the housing shortage is pretty bad. They apply the homelessness regs very strictly. Here, if someone had consented to the family home being sold as part of a divorce settlement, there is a real risk that they would regard them as intentionally homeless and decline to help. If they hadn't consented, but the court had ordered the sale regardless, that would be ok.
Anyone with £15k would be told that they have enough money to go and rent somewhere privately!
In this area, emergency accommodation for homeless families is sometimes a room in a b&b initially, but families are often immediately placed in a flat or house that is used for long-term temporary accommodation. This could be a property that is leased from a private landlord. They try and get families with children out of b&b as soon as possible.
If they accept a duty under homelessness, they are only obliged to offer you one property. As long as the offer is reasonable, ie big enough and affordable, they have discharged their duty. Having to change the kids' schools, being too far from family etc does not make an offer unreasonable.
You might want to discuss this with the homelessness officer at your district/borough council before you exchange contracts on the house sale. Hopefully, they will be more flexible than the council in the area where I work.