Is there any public node url exists of nodeos and keosd for EOSIO mainnet and testnet? Using that URL will access all RPC methods of chain and wallet.
There are several public nodes, since this is a distributed project, there is no single point of failure or central web site that documents them. There are several places where you can get a list of such services and these lists are dynamic in nature. Hopefully in the future this will change less frequently but usually it depends on the top block producers. They should have at least one public facing url for such api calls. Here are some starting points for your search: