Hello Franks
I hope you managed to solve your problem by now, but in case you haven't or someone else has it. Here's how I solved it with a quick explanation, I hope it helps you.
1) you mentioned that your "Dubai" iPad came with an AT&T chip. That's great news, as that is the chip you want to have. It means that your iPad works on a GSM network and not a CDMA network as Verizon's. I believe that at this point CDMA only remains in use in the USA, basically by Verizon.
2) the SIM PIN refers to the SIM you actually have inserted, not the one you just took off. (Personally I felt like an idiot when this dawned on me)
(At first I thought it was the AT&T sim that was locked)
3) you will find the SIM PIN for your new SIM on the back of the card you punched it out from. Use PIN, I'm not sure if PIN2 works or not. I guess it would, but i believe it is to be used in case you have a double line. You also have the PUK numbers, which should help in case you've used up your 3 tries.
4) try that pin, or use the PUK and contact your provider to unlock THE CHIP, not the iPad. The iPad isn't locked, but i believe that this point has been previously and quite bluntly stated. The Verizon iPad isn't technically blocked, but good luck finding a CDMA carrier other than Verizon's.
5) finally, just for reference, when you insert a sim from a different country, make sure you get the SIM PIN. Also check out the available services for each country. Once your iPad/iPhone recognizes the Network it is now living in, It will give you the services apple has "allowed" for that particular country. I.e. if you bring an Uk factory unlocked iphone to the UAE and install an Etisalat sim in. Guess what, no FaceTime for you. It happened to me. (A lot of international carriers block WhatsApp or Viber as it cuts in into their other services.)
I hope this solves it, and that you can now happily use a Swedish provider, in your Dubai bought, American market iPad. (As it is intended to)
Bob the fisherman whilst initially well meaning, failed to understand your problem, as he may not be familiar with network connectivity in different countries, or the characteristics of the different vendors abroad. (Like a guy in the Dubai mall selling imported iPads from the US). No need to be rude either