Analyze Statement I: Hyperconjugation is a permanent effect.

Hyperconjugation is the delocalization of electrons of a $\sigma$-bond (usually $\alpha$-C–H or C–C bond) into an adjacent empty $p$-orbital, partially filled $p$-orbital, or $\pi$-orbital.

Since this electron delocalization exists in the ground state of the molecule whenever the required orbital arrangement is present, hyperconjugation is classified as a permanent effect (like inductive effect), not a temporary effect like electromeric effect.

Therefore, Statement I is true.

Analyze Statement II: Hyperconjugation in ethyl cation $\left( CH_3-\overset{+}{C}H_2 \right)$ involves the overlapping of $C_{sp^2}-H_{1s}$ bond with empty $2p$ orbital of other carbon.

In the ethyl cation:

• The positively charged carbon is $sp^2$ hybridized and has an empty unhybridized $2p$ orbital.
• Hyperconjugation occurs from the adjacent $\sigma(C-H)$ bond of the methyl carbon into this empty $2p$ orbital.

The donating bond is a $\sigma$ bond formed by overlap of:

• carbon orbital on the methyl carbon, which is $sp^3$ hybridized,
• hydrogen $1s$ orbital.

So the bond involved is $C_{sp^3}-H_{1s}$, not $C_{sp^2}-H_{1s}$.

Hence, Statement II is false.

Conclusion:

• Statement I: True
• Statement II: False

Therefore, the correct option is: $\boxed{\text{C}}$

Comparison with stored answer:

Stored correct answer = C

My derived answer = C

So, they agree.