engine noises
 

two noises i'm wondering about and a car i just bought used:

1. on cold starts, i'm sometimes hearing a brief high-pitched whirring noise about a second after the engine starts. lasts for about a second or two. sounds like the starter or a belt could make that noise, but the owner claimed the starter was replaced.

2. after turning off the car, the back exhaust/muffler system is loudly tinkling for several minutes before it cools. sounds as if little metal bugs are hopping around in there, but it's probably just the sound of the hot metal cooling. still, does that suggest a problem?