I think either you rest or not, it has nothing to do with your body ability to quickly recover or not. If you are too sick and your body too weak, you are unable to do your normal activity. In this case you need to rest.

It depends on how fast your body can produce antibody. This usually differ from time to time. Your body will keep the antibody formula but it will remember the recent one better, thus it will take longer time if your body is infected by same bacteria that attacked several years back. If new type/group of bacteria, they your body will struggle longer for formulation.

Pain killer like paracetamol makes you feel better by reducing body temp but it is not a cure. However, i think the effect helps you on positive thinking and perhaps fasten the recovery... perhaps.