If I am a "contractor", does that mean someone has hired me to fulfill a contract? If so, does that mean that person who has hired me is a contractee, or what legal word could be used to describe that?
I have never heard the word "contractee," and the current edition of the American Heritage Dictionary, which deals with US English but also includes some words characteristic of UK or other usage, doesn't list it.
People who have agreed upon something in a signed document (contract) that covers their mutual obligations, of whatever kind, are called
parties to the contract.
In the US (and I have just checked a UK dictionary; what I am about to say appears to be true there, as well), a
contractor is someone who does work related to building construction or renovation. The word typically refers to a company or to an individual who operates as a company. A contractor usually performs the work under the terms of a contract with the owner of the property.