If I’m given resources by a client, should I look up the licenses of each one of those resources?

No. You should have written in your contract that client has all rights to materials he gives to you. With your invoice you should give him list (names) of files you bought to use for the project and the ones he gave you.
Sometimes I print dummy layout with signature so I will know which file is which.

I keep them for anyone trying to ask for license or extortion.