What's the difference between frontier and border?

To be honest I had to look this up just now!

Frontier is one of those words that I've always understood when it was said but didn't have an exact definition of it.

Here's a definition I looked up for frontier:
1.
a. An international border.
b. The area along an international border.
2. A region just beyond or at the edge of a settled area.
3. An undeveloped area or field for discovery or research: theories on the frontier of astrophysics.

I think that a border is talking about the precise "edge" of something (like a country), and frontier can be talking about the region near that edge. According to the dictionary entry it seems that a frontier can also refer to the "precise edge" of a country.

I think that all three of your example sentences are correct.

They're synonyms; however, "border" is much more common, especially in everyday speech. "Frontier" is mainly used in figuratively, or in the sense of newly-discovered land or new ideas.