B737 First Officer reported receiving a terrain and sink rate GPWS warnings which led to an unstable approach followed by a long landing.

Narrative

While descending into ZZZ on the arrival; for the RNAV Runway XX; we were twice placed in close proximity to preceding aircraft; leading to a 500 ft. - 700 ft. longer than desired landing. First approach didn't tell us we were following an aircraft on the arrival; until we were told to slow because we were overtaking them. The call and the slowing happened just as we passed the IAF; which led to us starting the descent late and having to descend at a higher than normal rate to catch our approach path. We did manage to get on the approach and achieve a stable approach by 1000 ft. Then; shortly after getting settled; the tower cleared a Company aircraft to depart before our arrival. They were told to expedite; but didn't seem to hurry. The other aircraft distracted me; due to the fact that I thought we likely would need to execute a go around. As we got closer; we were concerned that a go around would put us into a conflict with the departing aircraft. We did receive a couple of 'terrain' and 'sink rate' calls; but with the city lights and our position on short final; we both thought the calls were erroneous. Again; with the distraction of the closure rate with the preceding aircraft; and the GPWS noise; I allowed the speed to increase on short final. We did touch down a little long; but didn't want to go around due to proximity of the other aircraft. We both thought that it was safer to continue the landing than to go around.