The shifting, 1-3, and sometimes 3-5 is normal, as well as skipping 2nd on downshifting. The 6 speed would do the same downshifting. Sport mode, shifting is different as you found out. The difference between D and Sport mode sounds normal to me.

 

I've noticed the pulsing issue on a lot of ford trucks. Like you said, it seems to only happen with foot on the brake and the truck barely moving. With the truck stopped, whether in park or drive you can't feel it. Neither can you feel it with foot off the brake and the truck rolling. Not sure why, but it's always seemed to be as if the motor output was pulsing even though you don't see it on the tachometer or even hear it. Different generation, but I had a 1991 F150 that would do it really bad. I always thought it was uneven fuel mixture or fuel delivery based on the batch bank fired injectors. But I've noticed it in most ford v8's and even many of their v6 models. My own 2.7 eco does it as well.