Nobody has ever been able to remain in the UK just because they had a cat. The case Theresa May was citing was one where the person made an application for leave on the basis with his partner. There was a rule at the time that if a non-Brit had lived in the UK with their British partner (or it might have been spouse, I forget) for 2 years without enforcement action, they then had the right to stay. This policy is no longer in force. But it was when he made the application. The Home Office mistakenly failed to apply their own policy, which is not particularly unusual for them, so he was successful on appeal. It wasn't because of some kind of technicality, it was the application of the law. Nor was it because of the cat.
You would not be able to remain in the UK simply because you had a cat (trust me, I know of a lawyer who tried this argument and got told to piss off by the judge!). It's a bit worrying that May is either so ignorant of the ECHR and HRA, or is deliberately lying about it. Neither of those is a good thing!
An abusive parent also would not be able to keep the children simply because of the Article 8 right to respect for private and family life. That right isn't absolute anyway. The issue of whether parents right to family life can lawfully be interfered with by removing their children has long since been settled. It can be, and sometimes it. I certainly don't agree with a lot of the care proceedings system, but the faults in it are VERY DEFINITELY NOT because of human rights law.
Lastly, foreign criminals can be removed from the UK even if they have a partner and children. It happens. Judges and the Home Office are allowed to disregard the interests of British children if the offence is serious enough. There was a case this year called Lee where it happened, can't seem to find the reference though.