what are some applications for a 6'8" heavy/fast casting rod?

Depending upon the action, it's good for all the things that any heavy power rod is good for. Where a 6'8" rod benefits is for those who may be limited on space for traveling. Whether you're a bank fisherman who has to stuff your rods inside your car, or if your boat has a small rod locker and cannot hold a 7' rod. 

 

With all that, if it's a fast action, I'd have no problem using it for jigs, texas rigs, topwaters, swim jigs, frogs, etc. 

A lever?

Without seeing it my guess would be that the grip is somewhat shorter than on a typical 7' so the "effective length " for lack of a better word is probably very close to a 7'. It should do about anything appropriate for the power and action. Close work with a 3/4 oz spinner or deep vertical jigging come to mind

If the tip has any softness at all I'd use it for dock skipping jigs.  It would be good for pitching too, especially if you're in tight cover.

I have a 6'8" MH XF Cumara that is very stiff. I'd guess it is similar to some manufacturers' heavy. The rod is a great close quarters pitching rod with 1/2 ounce jigs and Texas rigs. 

I have a GLoomis with these specs, and it's my favorite rod for jigs around docks and lay downs, where you'll likely end up with a big fish on a short leash.

Docks and standing timber was the first 2 things that came to mind, but I have a 6' 9" H rod that I'm using for bladed jigs and really like it so far.

I have a H power Falcon Bucoo Micro that is similar (6'10" vs 6'8") and it is a great close quarters frog rod....IE skipping frogs under docks, and overhanging trees. Since I frog with straight braid, the micro guides (which I am generally not a fan of) don't impede me with the knot having to go through the guides. I was amazed at how much less effort it took to skip and the increased accuracy I gained with the micros, that my previous hatred of micro guides has been tempered a little.  It also works very well for pitchng jigs and t-rigged plastics under docks. I tend to keep the fluoro. leader short when using this rod for that to keep the knot out of the guides.

Close quarters combat when pitching jigs and plastics at specific targets.