First off, you can’t ship your car with the mattress/box spring, or vice versa (if that's what you're asking). They’ll need to be two different shipments.
For your car, contact car movers and get some quotes. This article will take you through some of the basics on transporting a car. It’s a pretty straight-forward process, but be sure and shop around for price quotes and make sure you have all of the details in writing (they’ll love the fact that you don’t have a strict timeline, that’s usually one of the biggest challenges of shipping a car since the auto shippers like to have a full load before moving the car).
As far as your mattress and box spring, it’s too small for moving companies to bother with. Check the traditional shipping companies like UPS and the like, and also companies that specialize in small moves, like MiniMoves.com. Another option is uship.com, which is like eBay for moving companies – companies go on there and see if there’s a load going their way. If they have room, they’ll bid to take it – it’s good for them because they get revenue and they get to fill their truck.
Whichever route you go, shipping the mattress/box spring likely won’t be cheap. In fact, you'll be required to put the mattress and box spring into moving boxes -- the boxes for these alone could run you $50 to $100. You can usually buy those cartons from a local moving company. Good luck with your move,
You should do the calculations and see if selling it in Miami and then buying new (or used) in Minneapolis makes sense. You might just come out ahead with this option.
Tim J. @ MovingAnswers.org Follow us on Twitter for valuable tips on making your move easier!