You started with "a dog is", so there is "is" for the cat part missing. The sentence after "and" does not contain a verb.
"There is a dog […] and a cat […]." works, as it is no more two sentences, but a list (a dog, a cat,… )
Does it mean that in Esperanto one may use the verb esti without a person of the verb and then it means: there is/ are?
What about other verbs: exp:
kreskas plantoj kaj bestoj. How would I translate that into English?