We have a 12 hour wait at Attaturk to connect to Kuwait. Can we get our checked luggage WITHOUT going through immigration? If not, do we have to pay the $60 each for a visa, just so that we can check-in our luggage again for our flight to Kuwait? Our flight itinerary from Canada ends in Istanbul and we have to do an additional check-in of our luggage with the Kuwaiti airline.....

Which airlines are you using? They may have an interline agreement, whereby the bags will go through to your final destination.
However, many airports offer support services airside to assist passengers with needs such as yours, so that you don't have to go through immigration. I don't know what the policies are at Attaturk, but you could ring: http://www.ataturkairport.com/en-EN/airportguide/Pages/information.aspx

The following was copied from the Atatürk Airport website:
"Transfer Passengers:(International Transfer Passengers With An International Connection Flight) A boarding card is issued for the passenger (if not issued in the country of origin) at the transit desk of the relevant handling company on the Arrivals floor. The passenger does not go through passport control. Instead, he/she directly proceeds through the transit area into the International Departures area. Baggage will be automatically transferred to the connecting flight."
So no, you do not need to get a Turkish visa. If for some reason you have to personally check in your baggage with the Kuwaiti airline the baggage still won't appear at the normal baggage carousel.

The following was copied from the Atatürk Airport website:
Good find. Yes, that information is on this page:

Thanks for this! The BIG question remains: how will our baggage get "automatically transferred to the connecting flight" when they don't have luggage tags directing them to Kuwait? That is why we have to claim our luggage and check-in again with the Kuwaiti airline. Can we claim our luggage at a carousel in the transit area, without having to go through immigration

Sorry, but I cannot give a detailed answer to the BIG question as I've never faced that situation during an international transit at Atatürk Airport.
What airline for Kuwait? And what airline to Istanbul? Why are you certain that your baggage cannot be checked through (or at least tagged) for Kuwait? I have faced somewhat similar situations where baggage was tagged for the final destination but I had to pick up baggage at a special return and then drop off with the connecting airline.

The BIG question remains: how will our baggage get "automatically transferred to the connecting flight"
We have asked twice now for details of your carriers. Just tell us, because someone here may know about the interline arrangements.
Otherwise, the only way to answer to your 'big question' is to ring the airline that will carry you from Canada and ask about checking your baggage through to your final destination. Or ring Attaturk to ask what facility they have to collect and transfer your baggage for you. This is done at other airports, such as BKK, but I have had no reason to do that in IST.

Air Canada to Istanbul....end of routing on that ticket. Atlas Airways to Kuwait, and they do not have an Interline agreement with Air Caqnada

Air Canada to Istanbul, and the end of our routing on that ticket. Atlas Airways to Kuwait, and they do not have an interline agreement with Air Canada, so AC won't ticket all the way to Kuwait. That's why we have to claim our luggage and get new tags sending the bags to Kuwait. I am hoping that we can claim our bags in the transit area and check-in with Atlas Airways in the transit area. If not, it looks like we may have to pay $60 just to claim our luggage and wait for 14 hours at the airport. Nice cash cow for the Turkish government??
Lesson learned.....Book our flights straight through to our final destination with carriers that have interline agreements, rather than trying to save money with cheap rinky-dink airlines. We actually used points to get to Istanbul and then had to find a way to get to Kuwait....

Perhaps you are making a problem where none exists. I can't find an exact answer but I've found a few similar questions mentioned at Atatürk with the same vague answer and no resolution mentioned. In my mind that means "no problem" as otherwise they would complain...
Your baggage should be able to be tagged by AC for Kuwait no matter any agreement (or lack of such) with Atlasjet. Atlasjet is a significant airline in Turkey--definitely not "rinky dink".