Cribbage beginner "Go" questions
In order:

- What happens if the opponent can't lay a card to bring the score to either 31 or under 31?
 Nothing; that's normal. The Go stands.

- What happens if the opponent lays a card and brings the total to under 31?
 Nothing; that's normal again. The Go stands.

- What happens if the opponent lays a card and brings the total to 31?
 Opponent gets 2 for Thirty One instead of the already scored Go.

Alternately, think of it as getting a second point for the Thirty One being reached exactly.

- What would happen if for example, the player who previously called "Go" has no cards to put down to lead?
 As in any other circumstance, that's another Go and opponent can play any remaining cards in hand as desired to maximize pegging value.

- Or what would happen if for example, the player who previously called "Go" is the only player with cards left?
 Same as before, but reversed.

- What happens at this point, as opponent has two Qs left?
 Opponent plays Q(10), Q(20) for Pair (two points) and Last Card (one point, functionally identical to go but called differently.)

- At this point, the Opponent can't play either Q. So are any points given?
 This is the default situation. Opponent scores the Go, and starts off the next sequence once already played cards have been turned over.



To remember the correct technique note that the proper timing for scoring a Go is once neither player can play anymore cards. At this moment in time, the last player to have played one or more cards gets either two points (if the count is exactly 31) or one point (if the count is less than 31).

The American Cribbage Congress Tournament Rules hint, but don't quite make explicit, this interpretation (my emphasis):


From the Scoring Chart


From Rule 7.2 Scoring


  a) Points are earned during the play of the cards for the following single card or combination of consecutive cards played. The player playing the last card that forms a combination earns the points (see scoring chart in Rule 1.7):
  
  ...
  
  (7) a cumulative count of 31
  
  (8) the final card played, not making a cumulative count of 31
  
  b) ... When a player's opponent has no more cards or calls "go," the other player may play all playable cards in succession and announce the points scored before pegging the total at one time.



So it is the combination of all cards played in this series after the opponent has last played a card that scores either two for Thirty One or one for whichever of Go or Last Card applies.