Except for some old Hudsons, all automotive clutches are dry clutches. There should be no fluid anywhere near the clutch friction surfaces. So contaminated transmission fluid or even contaminated hydraulic clutch linkage fluid should not make the clutch slip unless it has leaked onto the clutch friction surfaces.
I don't know why a clutch would be more inclined to slip when it is cold. My experience has been that it is more likely to slip when hot.
Also I don't see how old hydraulic clutch fluid would cause a slip. If anything it would tend to keep the clutch from fully releasing. If the fluid is old it would not hurt to flush it along with the rest of the brake fluid. You may find that the brakes and clutch pedals become a bit more responsive, but I think it is a long shot to think it would cure a slipping problem.
