Spacetime is sometimes depicted as a 2 dimensional surface with pits
representing massive bodies which curve the surface. I assume that
non-accelerating clocks on the flat surface of such a representation
experience no time dilation, and clocks sitting on curved regions do
experience some slowing.

(Please feel free to correct my understanding if I am wrong!)

My real question is this: if a clock was placed at the bottom of one of
the pits produced by a massive body where the surface is essentially
flat, would it run slow or would it run at the same speed as a clock on
the flat surface?

An associated question would be: is there an association between time
dilation and tidal forces such that a clock will only run slow if there
are tidal forces (assuming that the clock is not accelerating)?

Hopefully my questions are not completely nonsensical!

TIA,

E. Jones.