For the first question, she meant that she was happy that both of the sons were going to their own in-laws, that meaning that each son was going to the houses of the wives' families.
The phrase "out of your system" or "in your system" can refer to a lot of things, in the most literal sense things/substances you put in your body (such as food, vitamins, or even drugs) but here, the mother is using the phrase in a way to describe attitudes, feelings, or behaviors. In this case, she is not happy with how her sons behave, and she is hoping that going to their wives' families will help correct these behaviors/attitudes. In that sense, she hopes that the bad behaviors/attitudes will be "out of their system" which means that they will no longer act like how they used to.
Hope that helps somewhat!