Does RIP v1 routing protocol store "next hop" field in its routing tables despite not announcing it in the updates? And if yes, how does it do that, simply by storing the address of the node it received the update from or is there any other method used by the protocol?
A router's routing table is independent of the routing protocol, e.g. RIP, database. The routing table will have the next hop address. RIP stores the next hop in its database as the address of the router from which it received an advertisement. If that route is selected as the best route for the routing table, then the routing table will use that address as the next hop. RIP only knows about directly connected neighbors.