In a pusher configuration with turboprop(s), how is the superhot exhaust avoided?
Because the blade is moving in and out of the exhaust stream at high velocity, and the exhaust stream is perhaps 20% of the total exposure, the heat transfer to the blades is negligible from the standpoint of heating up aluminum enough to affect its heat treat, or heating epoxy to its transition temperature.  Meanwhile there are the anti-icing benefits.

There is an issue with the exhaust impinging on the prop on the Avanti, and there is a Hartzel Service Bulletin http://hartzellprop.com/wp-content/uploads/SB181A-R06-W.pdf that covers it.   The carbon and heat from the exhaust attacks the aluminum of the blade and the problem is blade corrosion once the paint finish starts to erode.  The SB is to inspect, clean up, and repaint the blades from this damage.  (Carbon and aluminum are at opposite ends of the galvanic scale and don't like to live together; this caused much grief on the CRJ program when someone who evidently missed their corrosion module in university decided to put carbon floor panels on aluminum support beams without an adequate barrier, with unfortunate results and driving a switch to titanium beams).

Really though, with the Avanti the biggest problem is the prop directly behind the gear acting like a FOD catcher for stuff thrown up by the tires.  You wouldn't want to try to fly an Avanti off a gravel strip, that's for sure.