Step 1: Evaluate statement I.
In eukaryotes, photosynthesis occurs in the chloroplasts, while breakdown of glucose (respiration) occurs partly in the cytoplasm (glycolysis) and partly in the mitochondria (Krebs cycle, ETC).
Thus, statement I is correct.
Step 2: Evaluate statement II.
Prokaryotes do not have chloroplasts. Photosynthesis in prokaryotes (like cyanobacteria) occurs in specialized membranes, not in chloroplasts.
So statement II is incorrect.
Step 3: Evaluate statement III.
Glycolysis occurs in all living organisms. It partially oxidizes glucose to pyruvate without oxygen. This is a universal pathway.
Hence, statement III is correct.
Step 4: Evaluate statement IV.
Glycolysis refers specifically to the breakdown of glucose, not glycerol. Glycerol metabolism follows a different pathway.
Thus, statement IV is incorrect.
Therefore, the correct choice is (I and III).
Final Answer:
\[
\boxed{\text{(A) I and III are correct}}
\]