Qualifying for Income Support means that you also qualify for Housing Benefit and Council Tax Benefit (your receipt of PIP care component also qualifies you for CTB).
Not wishing to pick holes, but can I clarify that bit which could be misunderstood -
Getting income support qualifies you for the maximum amount of housing / council tax benefit that you could get (this may not be your full rent if your rent is higher than the local housing allowance limit for private tenancies, or if you're hit by the 'bedroom tax' in a social tenancy.
You can still claim and receive housing benefit / council tax reduction on a low income, whether that income is from social security benefits, wages, notional income from savings (up to £ 16K) or any combination of the three.
You do not have to be on DWP benefits to get housing / council tax benefit.
Also, it's not an 'all or nothing' benefit - it's a calculation based on circumstances, income and rent. If your income is at or below income support level (whether that's IS or other income) then you would get maximum HB, if your income is a bit higher you would get most of your rent and so on, on a sliding scale.
Council tax reduction schemes work on a similar basis, although the rules are more localised now, and in some areas you still have to pay some council tax even if you're on IS level income.
You can even claim HB on a zero income, but they will want to know how you're meeting your living expenses (they tend not to believe people who claim to be existing on absolutely nothing)
If the OP is liable to pay rent and / or council tax and is not currently claiming HB / CTB, I'd advise claiming from local council soonest on the basis of current income.
As and when circumstances change, then report a change of circumstances - if you can't provide proof of new income immediately, this keeps your claim going and they will re-calculate it when you provide this (as long as you don't drag it out for ages)
Having a HB claim stopped because your DWP benefits have stopped, waiting for DWP to inform council (this can take time) and then doing something about it is not the right answer - there is a risk that your claim will be stopped from the date your DWP benefits stopped (and they will say they have overpaid you since then), and any new claim will start from such time as you do something about it.
It is in theory possible to claim 'backdating' (i.e. putting a claim in now and seeking HB for a period in the past as well as from now on) but this is difficult, and there has to be what's considered a very good reason - it's best to seek qualified advice here (CAB or some such.) It may be worth a go on grounds of health in these circumstances.
I'm not sure what the answer is to the bigger question - I'd be inclined to suggest re-apply for JSA - I think you're allowed to stay on JSA if you provide a sick note (or whatever they get called at the moment) for a while.
I would have thought that being refused ESA would be grounds for challenging a decision not to pay JSA.
I'm not sure that 'claim income support' is quite the answer here - income support is in effect the income based version of JSA / ESA and you need to meet the basic (unemployed / not able to work) criteria.
http://unemploymentmovement.com/forum/index may be worth a look as well.