Although "Has she got time?" is valid English and quite common in the US, "Does she have time?" is better English, even in America. It may very well be a sign of social status to use "have".
Let me put it this way: If you applying for a job selling cars, it's probably better to use "got" instead of "have" - unless the job is selling Mercedes, Rolls-Royce, Ferrari, or any other high-end vehicles. Then you'd want to use "have" - and you'd want to know why that makes a difference.
OTOH (On The Other Hand), if you're selling pickup trucks, you'd probably want to use "got".
It's not a judgment of anyone - it's just a description of common usage.
Duolingo has, up to now, neglected the formal, polite alternative for "tu", i.e. Lei, while in many other sources, including the Italian films I have watched, the formal address form is quite frequent. This question can indeed mean "do you have time", and I, for one, would not risk the "tu" form with people I am not close with..
'got' was used in Shakespearian times to mean 'begot', e.g. 'He got a son.' Thus a century ago, its use was frowned upon.
In my primary school, the word 'got' was written on a piece of paper, a hole was dug in the playground, and the word ceremoniously buried. The teacher reminded us to substitute a phrase with 'have' instead.
Hi HindiHaj - yes, it's true they can sound very similar and in a lot of cases the meaning won't help - "E un cane" (it is a dog) and "ha un cane" (she has a dog) are both grammatical. In real life, context might help. When it comes to the past tense, it seems you just have to learn which ones are conjugated with essere and which ones with avere , so you know it must be "E arrivato" and not "Ha arrivato" whether it sounds like it or not. Similarly, some phrases such as "ha paura" (she is afraid) use avere so it can't be "e" no matter what it sounds like. Please excuse lack of accents on "e" - I can't do them on this machine! Hope this helps.