What is the rattling noise I hear when I start my car?

Cars can get hypertension too, but in your case it's due to fluid overload, which often pushes things farther than they should go in a closed system causing damages such as blown out seals or loose rocker arms and springs. The latter is my best guess as to the source of your rattling noise. The springs should be tightened appropriately. (I tighten in the low valve position and then release a quarter turn).

Check the oil levels again. Does the noise stop right after it is started or continue then gradually get quite?