Historically, the threshold for "doing business" for purposes of jurisdiction was been very low. Any regular office or building used for the transaction of business would quality as would any work "directly" particularly at a state (e.g. if you did architectural work for a factory in Mississippi from your offices in Alaska, you would be "doing business" in Mississippi). Also, participation in a trade show would normally constitute "doing business" for a transaction arising out of the trade show.
A string of recent U.S. Supreme Court cases have tended to interpret this phrase more narrowly, but so long as a lawsuit has a nexus with the instance of "doing business" in a state that is in question in the lawsuit, the definition isn't that much different for most purposes.
A subsidiary or affiliate company's operations in a state will ordinarily not constitute doing business in a state on behalf of the parent company under recent U.S. Supreme Court precedents.
Suppose the company engaged a New York-based Plaintiff, and its
representatives signed the contract with the Plaintiff in New York
City. Does this satisfy the minimum presence rule?
Suppose, instead, the plaintiff and contract signing were in New
Jersey, but the company hired a law firm with offices in New York
City. Does this qualify?
This could depend upon the subject matter of the contract and where the negotiations took place.