OK, first of all, about PIP, it is basically for permanent conditions that affect your ability to look after yourself and get around. It's not connected to work
at all. I get PIP and I work full time. You might be eligible - it depends on your diagnosis and how it affects you - but the diagnosis does have to be for something that's not likely to stop in the next year or so.
Govt description that is pretty accurate
Universal Credit has a disability element. It's fairly easy to start a claim for it, and it gets them off your back for a while because they don't ask you to do anything to prove it until you're actually assessed, you just won't get the disability element until you've been assessed (and then it'll be backdated). I get a tiny, tiny amount of UC due to disability - my last assessment was years ago, but my condition is never going to change, and I think maybe they just can't be arsed reassessing me for the sake of pennies a week.
Universal credit is always better starting a claim for sooner rather than later, so it's good that you've started it now. They pay a month in arrears, and pay once a month. It's based on your net income (after tax, NI, and pension payments), so you'll need to upload your payslips. But it adjust very quickly if your income changes. So if you upload your payslips now, maybe you won't get anything this month, but as soon as you get your next payslips and upload them they will adjust your next payment to be higher. The online universal credit website is actually pretty good TBH.
You're not currently eligible for "new-style ESA." That specific phrasing is important. You aren't eligible right now because you're not unemployed - you're still technically employed, and you get Statutory Sick Pay. But it doesn't really matter, because you will be entitled to universal credit. All it means is that, for now at least, you don't need to think about ESA, JSA or anything else, just universal credit.
The govt link
Since you're getting SSP, that will be taken into account as income for UC, so the amount you get will be reduced. But that doesn't mean you get nothing even at basic rate, because you definitely will be eligible for "the housing element of universal credit."
When you apply for universal credit, you basically have to tick the box on the universal credit application form to confirm that you pay rent, and add all the details of your landlord and your rent. It will be a bit simpler with a housing association landlord than private, because they can check it so easily. It will be lower if you have a "spare room." It's 14% if you have one spare bedroom, so if your rent were £150 pw you'd only £129 (£150-£21) and would have to make up the rest out of your universal credit and SSP. Your council might have a facility to apply for a discretionary housing payment for the rest - you might or might not get that, but definitely won't if you don't apply.
Another govt link
That will also be paid in arrears. If you can pay your rent in advance for the next month, that'll help, but the HA really are not going to evict you for being one month behind due to UC payments coming in late. They might send you threatening emails and letters (my HA used to send me letters and emails every single week - and I wasn't even in debt, their systems just took too long to update! I eventually emailed over screenshots showing that - according to the rent account the HA maintain - I was in credit on the dates they claimed I was in debt, and asked them to stop harassing me, and they did), but they would never take you to court because no court would evict you for this cause.
I can understand you not wanting to go to the HA's benefit advice team if you have a spare room. Some HAs can be total arseholes. But if you contact them and tell them you're in the middle of applying for benefits due to being off work sick, they will take that into account when thinking about sending out threatening letters.
Council tax reduction has to be applied for separately. It might be worth phoning the council to ask for help doing that - they can actually be helpful, especially with someone in your situation, and they have teams whose entire job is helping people apply for it.
If you don't have enough to pay ahead a bit on your rent, you might end up a bit behind, but not so much that you'll be at risk of losing your home as long as you do star tgetting the rent paid. Honestly. It's a shitty, shitty system but it's not impossible to use.
In my area, there are benefits advice surgeries run at local libraries, and they might be easier to access than the CAB if your area has something similar. The CAB is very hit or miss, because it's volunteer run, so there might be someone really efficient and well-informed or you might, like I once did, get a trainee lawyer who meant well but made me sit there for half an hour while she confirmed what I'd said about PIP and ESA not being the same thing. So it's worth a try if need be, but don't give up if they can't help you, because they're not the only resource.
I'll help you with the actual application if it would make it easier for you. Anyone would find it hard, especially when they're already ill.