Yes it could but you wouldn't ever use the adjective 'braf' to describe a person.

Yes

A weather forecast would use this construction in the present tense. eg

'Yng Nghaerdydd mae hi'n braf y bore 'ma' = 'In Cardiff it is fine this morning'

But, yes in other circumstances this would be a sentence in the past tense = 'Roedd hi'n braf y bore 'ma'

I guess it's because the actual sentence is Mae hi's braf. The adverbial phrase just tells us when it was/is fine. And we know it can't mean "This morning is fine" because that's how I translated it, without thinking too much, and Duolingo rightly told me I was wrong. :)

That'd be:

Mae'r bore 'ma'n braf

"This morning is fine"

But here we have:

Mae hi'n braf y bore 'ma

"It's fine this morning"

The subjects come straight after the verb ("Mae").

When referring to weather (and time), we use the feminine "hi", yes.

It is a very common expression, in Britain at least.