From this thread:
Everything is explained on page 11, specifically:
Come into play: '..the effects are triggered when the card is played, either through normal recruitment process or through the mutation of another Goblin.'
Links: '..the effects are triggered whenever a card of the same clan comes into play either on its right, above it, on its left or below it.'
Swapping =/= coming into play, therefore swapping does not trigger links.
As an aside, you will never trigger links without having triggered and resolved another card's mutate (if applicable) and 'come into play' abilities first.
So the answer to all three of your questions is "No"; swapping does not trigger links, and link abilities trigger when the card comes into play, regardless of its final position.