Rats can produce on average, 11-18 babies per litter. Female rats come into heat approximately every 5 days. Within 24 hours of giving birth, rats can become pregnant again if the male is housed with her. It's always good to give female rats a few weeks 'rest' between litters to allow her to build up her fat reserves and fully recover. This period of rest should start after her current litter is weaned or otherwise...no longer there.
So,depending on whether you want to rest her or not, a female rat can have a little of babies once a month or once every other month with a resting period. Gestation is 23-25 days.
The best way to remove the stink is regular cleaning of cages, once or twice a week depending on size of the cage and how many adult rats are in it.