I haven't had issues using them as travel cables as they pack away really neatly, but I wouldn't rely on them for permanent patches because of their construction and susceptibility to noise.

You can buy cables that are claiming category 6 that are flat... But I highly doubt that they would pass a cable test (especially at that length) if it's in a noisey electrical environment.

You are right on your original assumptions, when you increase cable category for UTP, you increase the twists per linear feet. Cables can always claim they are rated for the category, but it doesnt mean they will pass a Fluke test or anything similar. You get a lot of cheap cables out there that may seem to work and light up gig-e connections, but then you get some weird connectivity issues every once and a while. It's why a lot of bigger contracts will ensure that all horizontal cabling is tested and certified after installation for the whole channel.