To determine the validity of the given statements, we must analyze the structural properties of the molecules described.
For Statement I, we look at 2,6-diethylcyclohexanone and 6-methyl-2-n-propylcyclohexanone. Both molecules share the same chemical formula, $C_{10}H_{18}O$. Metamers are isomers that have the same functional group but differ in the arrangement of carbon chains on either side of it. In these cyclic ketones, the 'sides' of the carbonyl group are the $\alpha$-carbons of the ring. In the first molecule, we have an ethyl group on each side. In the second, we have a methyl group on one side and a propyl group on the other. This variation in alkyl group distribution makes them metamers. Hence, Statement I is correct.
For Statement II, we consider the ability of 2,2,6,6-tetramethylcyclohexanone to undergo keto-enol tautomerism. Tautomerization involves the shift of a proton from an $\alpha$-carbon to the carbonyl oxygen. Looking at the structure, the carbons at positions 2 and 6 are the alpha carbons. Both are fully substituted with methyl groups, meaning they are quaternary carbons and do not have any hydrogen atoms attached to them. Without $\alpha$-hydrogens, the structural transformation to an enol is impossible. Consequently, this compound does not exhibit keto-enol tautomerism, making Statement II incorrect.
Final result: Statement I is true, and Statement II is false. This corresponds to the third option.