Well, that would be a round hoop because the round hoop provides a strong gripping pressure all the way around 360 degrees. Rectangular hoops grip very well at the corners but not nearly so well along the sides - so you need to do most of the fabric tensioning at the corners.
Having said that, if you are running a big design with lots of stitches you will find it quite a challenge to stop the fabric from becoming loose along the straight edges of a rectangular hoop.
You can try wrapping a strip of coarse fabric around the straight sides of the hoop to provide a little extra grip or you may have to use better quality backings or perhaps even an iron-on backing.
Barudan hoops have extra deep sides that are slightly concave so they hold together better on thicker fabrics.
If a standard hoop just doesn't work for you then you might like to consider the Mighty Hoops that are held together by super strong magnets. They are fantastic and provide another benefit as well. Because they have an upper and lower hoop surface rather than inner/outer, they don't leave marks on the fabric.
Yes, there are a couple of options now that work better on some machines than on others. The Barudan cap frame has an optional attachment that allows pocket embroidery.
In order to embroider on a pocket you have to be able to slide the pocket over the cylinder bed of the machine - so the bed is actually inside the pocket.
If your make or model of machine has a large cylinder bed, or if the pocket is very narrow then there won't be very much room for the pocket to move from side-to-side so you will be restricted to a very small logo.
This is more a matter of correct backing than hoop type. Due to its high stretch nature this type of fabric is best embroidered using a stick down backing like Filmolastic, which holds the fabric firmly in place during embroidery.