Warning: Not a lawyer or expert

I would double check with a housing authority under the assumption you signed a lease with an illegally large increase that you wouldn't be held to paying it after you've signed (that is that the rent increase above the amount legally allowed couldn't be taken from you despite you signing it and that you wouldn't be obliged to pay beyond that legal maximum).

If that is correct you can sign it, then just pay whatever the maximum allowed increase amount would be, if they make a fuss, sue and take them up with any housing authorities. They can't enforce an illegal contract (and I would think they also couldn't successfully evict you for non-payment of the illegal extra amount) but them having extended an agreement to you despite the illegal payment terms should mean they should lose, should have to let you keep the lease for at least the length of it (as any attempts to weasel out would I think be evidence of retaliation for ratting them out for the illegal contract and they clearly had no issues with you otherwise in offering you the contract) with only the legally allowed maximum rent being charged for the duration.

I would try and plan on having a place to move to by the time the lease is up though as there would be nothing forcing them to let you stay there after that and most landlords would just dump the old tenants, take it off the market for a month, then re-list at the higher price. And they certainly won't like you standing up for yourself which is another reason to be prepared after the end of lease date.

Don't sign this without researching that it is indeed illegal though for your specific circumstances. Some places have different rules for different types of housing like apartments versus single family homes versus section 8 housing so caps on increases may apply to one and not others. IF that is the case though you can sign without fear as they can't enforce an illegal contract and if they tried to evict you could appeal and show you paid the legally required maximum and that they tried to break the law and you should get a ruling in your favor.

As to whether they can kick you out. If you're not in a lease the answer is yes. For any or no reason. Many places have rules about a certain notice and grace period often depending on tenancy length (for example a longer length if you've been there for over a year but a shorter notice period if you've been there 6 months) but that's just a notice, maybe a cushion of 30-90 days if you're lucky though you are responsible for paying the rent they may choose to charge (the legal maximum they choose to charge anyways) during that time.

Of course this is a lot more work than just confronting them and not signing it but you risk them kicking you out whereas with this you likely are secure for a bit even with a furious land-parasite. Decide whether you'd rather spend your time playing these legal and regulatory games with them and staying on a bit longer or moving.