What are the realistic system requirements to run bitcoin core for generating wallets, signing and verifying messages on a laptop? I need to be able to verify messages and check that funds in wallets are valid and that the claimed owner possesses the private keys (hence checking messages signed). 

I know that I need to be able to download and sync the whole blockchain so I will be getting at least a 1TB HDD. 

Are there any other things that would be really needed to run it smoothly? I was thinking of a laptop with an i7-6700HQ processor and 16GB of ram.

Will this cut it?
You can just use the pruning, which lowers the required storage size to 550 MB. Other than that, you don't need a strong computer. It can run on a Raspberry Pi 3 B+ without any problems (but it won't be able to keep a full mempool).