Learn English – ny difference of meaning between ‘I am happy to be going home’ and ‘I am happy to go home’

This is a pretty interesting question. The answer is that there is a difference, but it's kind of subtle.

What they have in common: If someone just isn't present, they are both not here and not around.

Difference: "not here" says that she isn't here, in the room, right at this moment. "not around" is a statement that implies that she will not be around in the future. (Super technical note: That's a simplification. "She's not around" implies that she won't be here in the future relative to the context of the question; see my examples below)

If someone without a mother (either because the mother is dead or moved away) were to be asked "Where's your mother?", a common response would be "She's not around anymore." This automatically gives the idea that she's out of the picture for good.

"We could get your friend's help on this problem. Is he here?" 1) "He's not here." (He's not here -- but we might be able to get him.) 2) "He's not around." (He's not here, and we won't be able to get him for help on this problem)

They mean essentially the same thing: Are you going home this summer? which is probably the most neutral form of this question.

However Will you go home this summer? might be seen as being curt and could imply that you wish for that person to leave. It could essentially be taken as Will you please go home this summer? but without the please.

Will you be going home this summer? is more inquisitive. It implies that you are merely curious about to their summer plans and you don't particularly care what the answer is.

If this is being addressed to a person that you do not wish to go home for summer you could try asking You aren't going home for summer, are you? which retains the overall neutral value but implies that you prefer for them not to leave.