It's fairly problem free.
I just got back from Kurdish Iraq yesterday and the whole thing was fairly easy, though I think that that might change as they get themselves better organized.
The crossing from Silopi to Zakhu is the only safe one assuming you aren't coming from Iran.
I left Turkey and entered Iraq in less than 1/2 hr. and it didn't cost anything or require a Carnet. The 'problem' came with my bike- they had no clue really what to do since the only traffic they get local or long distance truckers. I got sent back and forth, waited around, and then I finally got one stamp on a piece of paper and all the others came in 5 min. So, after 1-1.5 hrs I was through to Kurdish Iraq and I didn't pay a dime.
I'll write a longer post about this, with more crossing details when I get a chance. It's a great place to visit!
- I saw people with Carnets, but I was never asked for one or anything. All the info I can get says that Kurdish Iraq does NOT require one.
-You get a free 10 days visa that can be extended at a central police station.
-They didn't ask for or look for bike insurance (and none of the security checkpoints after did either- about 5-10/day). I couldn't find any way to buy insurance, so I drove without. If you speak Kurdish or kind find a good english/your language speaker yu might have more luck.
-Getting back into to Turkey can be a bit of a hassle, but if you get stuck there are fixers who speak some English and other languages that can help. I ended up using one after I got sent between two different stations for a while. Cost 10TL (about 7USD) and got me through fairly quickly.