Hey man I am not 100% sure but like 99% sure. Only you are able to fast travel to your house.

Only you can travel to your house. Same way only you can travel to your inn location. The only time a group can transport to somewhere else is being called for an expedition or war.

Groups cannot travel to houses. It is player-specific. The cooldown can be reset for fast travels at these locations, and they work like inns, where weight is not a factor. The reset costs azoth (most times no more than 40 azoth to reset, but it depends on how much time is left on the cd).