If landlord puts rent up by £100 but I keep paying same amount I do now would that count as me not paying my rent (and could get me evicted)?
The LL could get you evicted for it, definitely yes. He's allowed to choose the rent, within reason. There are laws about what rent is allowed to be and if he hasn't put it up in so long and is still not putting it up to market rate now, then I very much doubt he is breaking those rules. You could challenge the rent raise through the fair rents scheme (or whatever it's called) but obviously that's going to piss your LL off, so as you currently have a good relationship it might not be worth it. Perhaps there's a helpline where you could ask someone what the likely outcome would be before deciding whether to raise a complaint? IDK.
Whether the council considers you voluntarily homeless or not, if you're paying the old rent but not the new raised part, IDK. Maybe it would depend on your finances as PP says and whether you really can't afford the raise, as opposed to don't want to pay it. That's what makes the most sense, otherwise everyone could claim not to afford their rent. You might be able to Google search the law/council rules around it, it'll probably mean wading through pages and pages of a PDF document to get to the bit you want.
As I mentioned before though, you're on sticky ground claiming you can't afford it when it's within the LHA rates for your area and is possibly cheaper than a council place too. HAs can evict for rent arrears too and they're going to reject you for the properties you want on affordability grounds, if you're openly admitting you can't afford the rent.
If your argument for not paying increased rent is "I'd rather have a council house for that price" then you're admitting to making yourself voluntarily homeless, aren't you? IDK the ins and outs of your circumstances but if you want help off the council you need to demonstrate a need for that help. Being fed up of paying private rents or wanting a garden isn't a need, it's a want.
The charity Shelter could maybe answer some of your queries about rent/LL/eviction/the law. They operate from a perspective of "how not to be evicted" though, not from a perspective of how to obtain a council property.
There's nothing wrong with you wanting what you want just so long as you don't actually expect it to happen. If you're pinning your hopes and plans on a 3bed house with garden from the council, you're going to be chronically disappointed and unhappy, which doesn't help your quality of life at all.
Council properties don't exist just to house people who don't want to pay private rents, there's just not enough of them to go around. The majority in private rents don't want to be there, a lot would rather be in council home or buy somewhere.
There's nothing to stop you from continuing to bid on what you want and maybe you'll get lucky one day, you never know. If you do decide to use the homeless scheme and get given a permanent property you hate, there's also nothing to stop you giving it up and going back to private rental to get a house with garden, if that's what you want.
If you had to choose or die, what would you pick - a 2bed terrace/flat without garden from the council or a 3bed with garden from a private LL? (Going to assume you don't pick death!). Because if you'd rather have any council property than no council property and if you could manage in a 2bed, you might be more likely to get somewhere sooner by bidding on those, than by bidding on the 3bed semi with a private garden that the world and his brother is also bidding on. Same with checking the banding of properties before you bid, if you're bidding on the ones in your banding you're more likely to get them. Basically what I'm saying is, if you really want a council property and you want it sooner than later, then don't waste your bids on unrealistic dreams.
I know life can be shit but there's no point doing that old person thing of being shocked at the price of eg bread because "back in my day it was tuppence ha'penny and now it's £1.50!!!", if all the loaves of bread are £1.50 and have been for some time. Moaning won't change anything and if they want bread they have to pay the price and that's that.
I dread what's going to happen if the CoL keeps rising and more and more people can't afford to survive. I can see there being mass homelessness and people living in slum tent cities like a 3rd world country.