Does shifting into Drive or Reverse after a cold start cause transmission problems?

Definitely, and what you’re doing is exercising what’s called mechanical sympathy, and it’s a dying art, I’m afraid.

When you consider the forces involved in shifting a car from Neutral to Drive or Reverse, it’s easy to understand that doing so while the engine is revving higher than idle is asking for trouble. Extra revs mean extra forces on all those hydraulic and mechanical bits that make up an automatic transmission. It’s a bit like the difference between taking off normally in a manual car versus revving the engine up and dumping the clutch to make a fast getaway. Yes, it will leave the green light faster, but you’ve just multiplied all the forces trying to tear the transmission apart.

A slightly higher idle shouldn’t make much difference, but the golden rule is to only ever select a gear when the engine is idling at its normal rpm. The good news is that modern engine management means a cold-started engine should only idle higher for handful of seconds, meaning you don’t have to wait long before selecting your gear and taking off.

By the way, a healthy transmission shouldn’t clunk into gear either; the gear-selection process should be swift and all but silent.