We have been lucky enough to be flag family twice now. It was a unique experience. Both time we were able to do it were on a 9 night trip and even then it was only at the very end of our stay. Our last trip in Feb. I asked about the flag family and the CM said, "I'll see if they have checked in yet, are they at the lodge or the villas?" I then explained about the flag family program, and she had never heard of it(she was new). We were eventually able to sign up!
About the carriage or hay rides, they are offered over at the Fort Wilderness campground. The carriage ride is $30 for a half-hour ride. It can fit up to 4 adults, or the driver said 2 adults and 3 kids. You can make ressies by calling 407-WDW-PLAY. The carriage ride was fabulous, we have ressies for another in July. I would reccomend calling, the evening we went on the ride, I heard 3 couples ask the driver, and he stated he had no openings left. The hay rides are, I think, $7 for adults and $4 for kids and is a 45 minute ride. You cannot make ressies for those, I was told it is first come- first served. We have not done a hay ride yet, but hope to soon. Have Fun!