Translation:They have a cat which has many toys.
Not only is "that" a viable alternative for "which" even though it's rejected, but the comma doesn't belong in the English sentence because the relative clause is restrictive. Edit: On second thought, this is one of those cases where the relative clause can be both restrictive and non-restrictive.
I interpret that slightly differently. I would say that the comma is optional in the English sentence and actually changes the meaning slightly. "They have a cat which has many toys." (no comma) means that the cat and the toys are pretty much inseperable and the toys are an important part of how you think about the cat. "They have a cat, which has many toys." (with comma) means that they have a cat, and, by the way, the cat has many toys, but they're not really important.
The subject in "jolla on monta lelua" is "monta lelua". It's not the owner but rather the owned thing that is the subject in Finnish ownership clauses. The owner must in adessive case, which as a locative case cannot be attached to a subject because it expresses location and therefore functions as an adverbial. This also means that the subject in "heillä on kissa" is "kissa".