Learn English – Where does “give something a whirl” come from

In live hare coursing, a bye means to pass to the next round due to a missing opponent, but not before the dog has made a lone run so it doesn't have a rest and an unfair advantage over the others.

The National Coursing Club's glossary says:

Sometimes runners are withdrawn from their courses, either because of absence, injury or weariness. Their opponent still has to run a course – a “bye” – so that it will have run the same number of courses as the next opponent. The dog may run alone or accompanied.

So in the first example above, greyhound Bismarck's opponent had pulled out, meaning he could progress to the next race, but not before having to run a punishing run to "even the playing field".

And in the second example, the day ends with greyhound Duffer running a bye on its own.

And again in the third example, where it's clear the opponent was injured the day before.

We can see bye used a lot in any of these books, and "run [...] bye" in many other coursing books. As can be seen, "get a bye" isn't in itself any kind of special set phrase (at least in these examples), but just some words used together.

Swag as a dictionary entry traditionally has several meanings, including the one you're referencing.

Swag as a slang word has its roots in "swagger".

Swagger: A very confident and typically arrogant or aggressive gait or manner

It refers to a way your present yourself; having style, confidence, etc. Using the term is so popular among young people that it's generated quite a bit of backlash, especially among the internet culture. "Secretly We Are Gay" and the like are simply backronyms that serve to make fun of people.