Valve guides or seals could be worn allowing oil to build up and be sucked in to combustion chamber
no adjustment on hydraulic clutch, worn clutch disc and pressure plate
With fluid. There might be a bleeder scerw on the slave clinder.
If it is on going from neutral to any forward gear, bleed the lower resovoir. On sitting for a winter of even a month on hot dry weather will cause air to leak in. If it is on fast shift to third the fluid return hose has a metering block in it to slow down shifts. This can be reamed out. After a fast first to second the second to third doesn't have enough fluid back in the upper resovour and locks you out. A good Corvette mechanic could help you here or look to the internet and somewhere the ream out is detailed. While you are at it use a turkey baster and pump out the upper and depress pedal and give it 3 mins to fill then re pump out with baster. Use STP Synthetic power steering fluid DOT 4 or better.
no a grinding problem is usually in side the trans and what is grinding are the gears...there are brass bushing called sincros that help transfer from gear to gear...since the "sincros" are brass they wear and when they wear there isn't the smooth transaction from gear to gear...sincros are really cheap but to have them installed gets pricey because of the labor time..