Let's say I have the geometry of the rotor: the number of blades, the airfoils that make up each blade, their position, chord length and their twist angle. Would it be possible to use this information to determine the tip-speed ratio?
I would point you to the 2014 paper by M. Ragheb titled Optimal Tip Speed Ratio http://mragheb.com/NPRE%20475%20Wind%20Power%20Systems/Optimal%20Rotor%20Tip%20Speed%20Ratio.pdf which includes a fairly clear description of the factors affecting turbine efficency.
For a rotor with a variable pitch system, the rotational speed vs. power generated can be controlled by feathering the blades to accommodate varying wind speeds and therefore keep the system operating at the best TSR.
For a fixed pitch system, some method of governing the maximum RPM and will be required.