Learn English – What does “we’re still here” mean in this context

The verb "dog" has two possible meanings:

1. dog somebody/something

   (of a problem or bad luck) to cause you trouble for a long time

   Her career was dogged by misfortune.

2. dog somebody/something

   to follow somebody closely

   She had the impression that someone was dogging her steps.

Since the subject of the sentence is "Disagreements over the timetable," we can safely assume that the first definition applies, since a disagreement is a kind of problem:

Disagreements over the timetable... caused long-term trouble for Mr Obama's relations with...

In this case, aye almost certainly does mean always. For one thing it makes perfect sense contextually. Moreover, the connection with Scottish actually makes a lot of sense. We learn from Morag, in another answer, that aye is still used in Scotland. Even if it weren't, archaic English or Scottish would not be out of place in this song.

The Watson Family lived in the Appalachian Mountains, a region settled by Scottish and Irish immigrants. Difficulty of travel kept them functionally isolated, and the pre-radio music of that region is known for strong ties to old ballads of Scotland, England and Ireland. Here's a great radio piece about this history: http://hereandnow.wbur.org/2014/10/01/wayfaring-strangers-book

The corpus of Appalachian folk music also includes a lot of references to political roles that never really existed in the U.S. - Lords, ladies, knights, kings, etc. Examples abound here: http://www.contemplator.com/america/index.html