Big tires are rubbing in certain conditions so how can I fix my rear wheel play issues? I have a 2007 Tarmac Expert Double with a 
Mavic Kysrium Equipe rim spoked to a Mavic Kysrium Equipe, 24 hole, Mavic QR rear hub.

I recently upgraded to 28mm tires from 25mm and I had the rear wheel recently trued so rolls smoothly and clears all the stays.

However, when I climb (out of seat) or make sharp turns, the tire rubs on the sides of the seat stay. 

When I press on my rear wheel to the left or right, it gives enough to rub against the brake pad as well as the sides of the seat stay. If I loosen the brakes the tire rubs a bit more on the right side than the left.

Gooling around shows that my hub might be going bad given that I have put about 5-7k miles on the bike.

Is this repairable? Does it make sense to buy a new wheel? Can I just replace the hub with a PowerTap and new spokes? How dangerous is it to ride with so much play with the wheel.
Unfortunately 28mm nominal tire width can be too larger for many modern road bikes. Note that actual measured with depends on many factors including: internal rim width, tire construction, tire profile, tire age and wear, inflation pressure. As such, one combination of tires and wheels can work while another may fail if you have tight clearances.

It is also impossible to build a bicycle and wheel set without flex.  As such, there needs to be a minimum safe tire clearance. Just because the wheel spins freely in the stand does not mean there will be appropriate clearance for actually riding the bike, especially for heavier/stronger riders.  I would suggest assessing how much clearance is available with 25 mm tires (perhaps(typical max clearance on many road bikes).  If the clearance is already tight (e.g., less than 1 cm) then going up to 28 could be a stretch. 

The more recent version of the tarmac (e.g., disc version) can clear 28 mm, but I doubt older rim brake versions can.  It is best to ask the Specialized Help Desk to get an official answer. 

Question of Hub Play

All wheels, will have some side-to-side flex so it is unclear whether or not there is a problem with the hub. Typically, if the bearings have developed play the wheel moves side-to-side with a very light touch, if more force is required it is likely wheel flex.  5-7000 miles is not a lot of mileage, but this also depends on the conditions they were ridden under.