Normally, when a corporation hires employees in a state where the corporation is not incorporated there is a high probability it will be considered to be doing business in the state where the employees live, even if it is not registered there. In fact, some states have laws that say that if a corporation does not properly register in a state but conducts business there, the corporation can be sued in that state BUT, the corporation has no right to defend itself in the courts of that state.
As you mention International Shoe, what has come down from that precedent has been the idea that a corporation's "minimum" contacts with a state, such as advertising in a state, selling in a state, hiring employees in that state, etc., will usually mean a company is considered to be doing business in that state and subject to jurisdiction in that state.
In addition to the situation with International Shoe, there is also the fact that when you have a corporation from one state and employees in another state, you may have a situation which involves the question of which state's laws will control the case. It is possible, for example, for a court in one state, like CA, to decide a case using another state's laws, like NY, if the state where the court sits' "conflict of laws rules" determine that another state's laws need to apply.
The legal ideas of jurisdiction, corporate registration, employment law, and proper tax payments are all part of your question. You truly need to meet with an attorney who can help the corporation figure out not only where it can be sued, but also under what laws, and whether it needs to register in another state as a "foreign corporation" (a corporation registered in NY would be considered a "foreign corporation" in CA), and what to do about such matters.
PLEASE NOTE: Responses here are for information/education only, NOT legal advice and do not form attorney-client relationship! Only licensed attorneys you hire in your state can provide legal advice.