If you examine or search something with a fine-tooth comb, you do it very thoroughly or carefully.
('Fine-toothed comb' is also used.) To search very carefully is to search with a fine-toothed comb.
If someone is angry or unsettled, telling them to cool their jets means they should calm down.
If you do something out of the kindess of your heart, you do because you are kind, not for any benefit or out of duty.
('Out of the kindness of your heart' is also used.)
This means that someone is very nervous or jumpy.