"Do you have cats?"
Translation:У тебя есть кошки?
"Кошка" is a female cat and "кот" is a male cat. When the gender is not specified, it's more common to hear "кошка" as a translation of "cat". So your translation is right, but the proposed one sounds more natural.
Instinctively I think to say У ты есть кошки, thinking you/your is ты , explain why this is wrong? please?
Is кошка irregular like девочка in that it adds an и instead of an ы to pluralise it?
This pattern is regular: consonants like г, к, х and ш, ж, щ, ч always use а, у (not я or ю) and и (not ы) right after. As a consequence, if a noun ends in -К or -КА in its dictionary form (e.g., мальчик, носок, флешка, кошка, девочка) it will use И instead of Ы in its case endings.