Is it possible for a Dapp that uses the metamask/chrome to set up a browser cookie? or can a cookie set by another website like facebook, read some information or at least know that a dapp has been visited ?
A dApp consists of two different parts: the blockchain part and the more traditional www interface part.
The blockchain part has no idea about cookies, it's deterministic and can't communicate directly with the external world. All information has to come and go through nodes.
The more traditional www interface part is almost just like any other website. So it can set cookies and whatnot. The only difference is that it also has a component (web3) which can connect to a blockchain node to talk with the blockchain. Metamask is one option on how to facilitate this connections.