No and yes in that order. The clock on the ground would apparently run slower than the clock on the tower. I'll do a thought experiment: Special relativity says that moving clocks appear to run slow and the faster they move the slower they appear to be. But General relativity says that special relativity only works relative to a freely falling reference point. ok, I'm getting an imaginary airplane and jumping out free fall over the tower with my watch. As I pass the clock on the top of the tower, i see it moving upwars relative to me at some rate and can work out how slow it seems to be relative to my watch,. Just before I go past the second clock and go splat I figure it seems to be moving upwards slightly quicker (because I'm now falling a little faster), so it must appear to be going slower by that much more than the first clock. Clocks are slowed by a really really small amount in the earth's gravitational field. Nevertheless the experiment was done with a 22.6 metre high tower in the 60s at Harvard and the a slow of five parts in 1,000,000,000,000,000 was detected just as predicted. EDIT: NOTE TO ANSWERBAG STAFF: This question has been moved to "clocks" but it is not a question about clocks, it fits better in "Gravity", "Relativity" or "Cosmology".

No. Watches measure time, but they are programmed or wound to keep time at a constant rate.