This works whenever you're on the defensive, i.e. NOT rolling the dice. So it'll work when someone attempts to grapple you, when someone attempts to maintain a grapple, or when they try to pin you.
It doesn't apply when you attempt to reverse or escape a grapple.
There is an answer in plain English and there is a loose definition based on the rules.
If you scroll all the way down, my answer for the Goliath is a No. Or at least not without him going through some more hoops.
The plain English answer
You are carrying whatever you and the GM agree you are carrying. For example, by the Merriam-Webster dictionary :
Yes, the goblin would teleport with him.
The magic item says that it brings along everything you're wearing or carrying, without specifying objects or creatures, so presumably it would refer to both, so that it can properly affect everything you're wearing or carrying.
This wouldn't let you bring someone along by holding their hand or grappling them, but if ...
My answer would be "No, but..."
No, the "using your attack action to grapple/shove instead" rules would not apply, since it's not a creature - you quoted the relevant rules already.
However, the list of actions one can take in combat is non-exhaustive, as per "Actions in Combat" in the SRD:
"When you describe an action not ...
Strictly RAW, It goes pretty much like Kirt says above.
Some ready their move, one does the moving, at half rate dragging the target, the others move along, maintaining their grapples. The target on his turn may try to break the grapples, and that would work like most such group activities -- the best grappler rolls with advantage since the others are ...
While I agree with the answer of MrNattious, consider that the OP asked for what happens when "multiple PCs grappling the same target NPC?" Often when this happens it is because the PC's are attempting to help one another move together, so I think it may be valuable to point out how the PC's can act so as not to "get left behind", if ...
Some yes, and some no.
Bigby's Hand and Unseen Servant are tangible -- they have AC, hp, Strength -- so I would allow a creature to grapple/shove those.
OTOH, Guardian of Faith is stated to be spectral, and has no physical stats, so I would say "no" to that one.
Spiritual Weapon is also stated to be spectral and has no physical stats, but also ...
No, grapple and shoving attacks can be performed on creatures only.
You quoted the relevant rules: both attacks require that the target be a creature.
In particular, about your listed entities:
Bigby's hand is an object, not a creature: it can grapple because its description says so, under the Specific beats general principle.
Guardian of Faith is neither a ...
Yes, it can be done if your DM is willing to allow it.
3 important things here
1. There is no air in the bag of holding unless it is open so an unconscious creature would average approximately 4 minutes to be revived from the inevitable suffocation because they cannot hold their breath.
2. The "inside" of the bag of holding is an extradimensional ...
Teleportation spells state when they permit a passenger.
Let's compare three spells.
[...] You can also teleport one willing creature of your size or smaller who is carrying gear up to its carrying capacity. [...]
Thunder step explicitly states that you may bring along another creature.
[...] You can also bring one willing ...