Step 1: Evaluate Statement I.
In protozoans, gliding movement is brought about by contractile protein fibrils called myonemes.
Hence,
\[
\boxed{\text{Statement I is true}.}
\]
Step 2: Evaluate Statement II.
Asexual reproduction in protozoans generally occurs under
\[
\boxed{\text{favourable conditions}.}
\]
Sexual reproduction is usually associated with unfavourable conditions.
Therefore,
\[
\boxed{\text{Statement II is false}.}
\]
Step 3: Choose the correct option.
Hence,
\[
\boxed{(C)}
\]
is the correct answer.