Capitalization – Should ‘A’ Be Capitalized at the Beginning of a Sentence?

Yes. Sentences start with capital letters; abbreviations are no exception.

A possible* exception is when a proper name starts with a lower case letter. E.g., if I changed my name to matthew then "matthew is awesome." would be correct. This is because the word is intended to be lower case. E.g., on the other hand, has no such association with it.

_{* Don't make an exception. This is just playing devil's advocate. See comments.}

Yes. In the Australian context “Aborigine” and “Aboriginal Australian” should appear capitalized because they are acting as proper nouns referring to specific group. When being used to refer to indigenous people in general they would not be capitalized. For example if you were to write:

This is a fine piece of aborigine art.

you would be talking about a piece of art created by the original inhabitants of their respective region. But if you were you to write:

This is a fine piece of Aborigine art.

You would be talking about an artwork from the Aboriginal Australian culture.

Thus one could write:

The aboriginal people of Australia are the Aborigine.

And, if—for some reason—you were to write:

This is a fine piece of Aborigines art.

You would be referring to the Roman myth.