I think têm is more colloquial. Há (inf. haver) comes from late latin and means also to have but is more used in grammatical context (like I have learned) whereas têm (inf. ter) means more to posess and then people forgot the difference and used both for there is.
têm with circumflex is for when the subject is plural, like eles têm um cachorro. In the sense of to exist the use of tem is informal and in the singular form, as the formal variant haver, so it doesn't have circumflex.
Since "na bolsa" invokes the idea of inside before anything else, we would use the alternatives shown by pfeil.
Literally 'Ela tem um gato/uma gata na bolsa dela', and nit-pickers could say that if she lent me the bag and a cat got in, 'There is' would apply and 'She has' wouldn't. Narrow case.
It's not right. Use "there is" for singular and "there are" for plural.