Do routers verify the actual IP packet size with that mentioned in the packet?
Routers should check the IP length, but are not required to. If they check, they are required to drop mismatches between the value in the IP header and the actual packet length. Routers should depend on link-layer payload detection, and compare with the IP header per RFC 1812.
This is covered in RFC 1812 - Requirements for IPv4 Routers, Section 5.2.2: IP Header Validation:
Additionally, the router SHOULD verify that the packet length reported by the Link Layer is at least as large as the IP total length recorded in the packet's IP header. If it appears that the packet has been truncated, the packet MUST be discarded, the error SHOULD be logged, and the router SHOULD respond with an ICMP Parameter Problem message whose pointer points at the IP total length field. DISCUSSION Because any higher layer protocol that concerns itself with data corruption will detect truncation of the packet data when it reaches its final destination, it is not absolutely necessary for routers to perform the check suggested above to maintain protocol correctness. However, by making this check a router can simplify considerably the task of determining which hop in the path is truncating the packets. It will also reduce the expenditure of resources down-stream from the router in that down-stream systems will not need to deal with the packet.