I'm guessing the "se" is the same reflexive "se" you find in most Romance languages, and "habet" would be the third person conjugation of "habere". Does this literally mean "Livia has herself fine"?
I am not sure but, in Spanish would be like: Livia se siente bien,
se=without translation; siente=feels=habet; bien=fine=bene.
It seems the difference in Latin is that adverb is before verb.
If that were the case, linguists would not have been able to conclude that it is fundamentally an SOV language.
Flexible syntax does not mean anything goes and it's total chaos. There is always a default/preferred/unmarked order.
Yes, word order is relatively free in Latin, but there is always something underlying why it was sequenced this way this time and that way that time. Generally it's something simple like emphasizing something. Of course, in lessons like what Duolingo provides, there is no context to promote one way over another, and this gives a bit of a misleading impression that the different ways of saying it are 100% interchangeable with no nuance of difference.
There really isn't. There is an expected word order, but it means the same thing. If you change the word order, it can be used to emphasize specific words.
As far as I know H was pronounced in classical latin, but mainly from the educated upper class. It had already started to fall out of use in everyday conversations back then. So I guess in duolingo-like not-particulartly-formal context, it might as well not be pronounced.
This course is meant to teach Classical Latin. Therefore you should expect the H to be pronounced.
"H" is sometimes pronounced in French. "aspirated h", and it's a Romance language too. Do you mean aspirated or something else?