Q: Will there be any gain in performance if MTU size is increased on NAS and switch port to which NAS is connected if the MTU size is not increased from the standard 1500 bytes on the client NICs?
Answer: No, because the the increased MTU size is not being utilized by the client. If you wanted to transport 100 people from point A to point B, you could use two busses or 25 sedans. If the road between A and B is made so that the busses can more easily move with no delays, and you still move them in sedans, you're not gaining anything.
Q: Will this cause fragmentation problems?
Answer: No, fragmentation happens in the opposite scenario when you send a large packet while the path does not support it and needs to chop it into supported sized packets.
Q: How do devices "handle" mismatches in MTU for interfaces across a switched network?
Answer: If the packet is smaller than the allowed packet size, it is passed with no issue. If the packet is larger than the allowed size, it is dropped.