Trying to kick a cold. Is it beneficial to go to gym and "sweat it out" on a bike and then steam room, or better to just rest and only rest? Thank you

Neck rule. General rule regarding exercise when ill: if all symptoms are above the neck with no fever and minimal fatigue, then you may exercise. However, if you have a fever, chest cough, shortness of breath or significant fatigue, then you should rest until improved.
Neither. When one has a minor respiratory infection, my advice is to continue exercising but to decrease the intensity of the work outs by 30-50%. Going "full bore" or doing nothing are both not as beneficial. See your doctor, however, to make sure a lighter work out is ok.
Either. If you feel well enough to exercise without it exhausting you, go ahead. Generally, however, rest is always good. Colds are due to viral infections, predominantly of the upper respiratory tract and may produce sinus and ear congestion. This may severely limit the amount of physical activity you are able to tolerate. You might want to use a decongestant before you begin.