Traditionally, the IBI has always been payable by whoever holds title to a property on January 1st of the calendar year in question - so if you own a property on Jan 1st, you are liable for the IBI whether you own the property for the whole year, or sell it, whether that be in January or December.
There was a High Court ruling last year I think, allowing for sellers to pass on a pro-rata amount of the IBI to the buyer, but as far as I know town halls will still hold the title holder as of Jan 1st liable, and therefore it's up to the seller to claim this proportion from the buyer by private agreement. Since like most things in Spain tradition usually holds sway over law, at least for some time, it's unlikely that the new ruling will be enforced in any meaningful way for the forseeable future. In the case in question here, it's highly unlikely that the seller is going to bother claiming approx. 1/12th of their IBI payment for 2017 from Catlady.
As for the tax office making claims for additional transmission tax, I've personally heard of cases where the buyer has had a tax demand within just a few months of purchasing, so I presume the 4 years mentioned refers to the statute of limitations - although I thought it was 5 years rather than 4. In my experience, when the tax office believe they are owed money they move fairly quickly; if they owe a rebate, it can take ages for the wheels of bureaucracy to turn!