"Mom does not have time for the dogs."
Translation:У мамы нет времени на собак.
It is the other way round, meaning-wise. Both are used anyway, and both are common (with the approrpiate case). For example, you can use "У меня нет времени на игры" or "... для игр" for "I do not have time for games", same with "на подготовку"/"для подтотовки" ("for preparation").
на is specifically when you are talking about the amount of time to be spent for some activity. This works in exactly the same way на is used with other resources to be spent on something.
(however, you can only say "Самое время для прогулки" to say it is a good time for a walk; "на прогулку" does not express this meaning)
In the standard language the Genitive of время is времени, not время. The use is basically like everywhere else: when there is "no" something, you use нет with the Genitive for of that noun.
However, you'll see нет время/ сколько время in some younger speakers, so this change might happen a few decades from now on. Or not. It hasn't happend just yet.
I thought на meant on why do words keep changing meaning? why is it мамы and not мама i thought adding ы on the end of a word made it plural and why is it собак and not собаки? All this time ive been taught собаки meant dogs and now im being told to use собак for dogs and instead of кошки i now use кошек for cats (or just кот sometimes) and its really confusing me