The information you have received is correct. In regard to suite awards, you won't find the three night minimum on the individual property websites, it is part of the Hyatt program, so it is on the program website. Suite awards do have to be booked by phone and can be capacity restricted, there is no guarantee that a suite available for cash will be available for points, like there is with standard rooms. This is also true of points suite upgrades on paid nights. The cash upgrade option is hotel specific, not a program feature.
I think that some people might use adjoining and connecting interchangeably, however, I am pretty sure adjoining just means that they are next to each other.
On another note, I am not sure how you might be able to make it happen because capacity is officially 3 for their rooms, but there is plenty of room for 4 in a standard room. We booked two rooms for the 6 of us, and just moved the other rollaway bed from our room to the kids room and the four of them slept in one room while my wife and I had the other.