It depends which native speakers you're talking about! I can honestly say that never heard anyone say 'brown-bag' before today. ( In fact, it sounds a bit unpleasant...it makes me think of 'brown-nosing' and similar expressions). In the UK, the usual expression is 'bring a packed lunch' or simply 'bring your own lunch'. I suspect that you would confuse most native speakers worldwide if you started talking about 'brown-bagging' to them. I presume that this expression is mainly used in the US. By the way, there's a small error in your sentence - it should say 'every day' as two separate words. This is an adverbial phrase. The word 'everyday' only exists as an adjective ( e.g. an everyday occurrence).

"Brown-bag" is a common colloquialism in American English for "to pack lunch" or "to bring one's own lunch", in contrast to buying lunch. The example sentence you used is correct!

Probably depends on which part of the world you're talking about. In Canada, in my experience, we'd most often say something like "I decided to make my lunch everyday (to save money)". To bring a lunch box is also fine of course, but it makes me think of children bringing lunch boxes to school. Brown-bag sounds familiar but I haven't heard it very often. I've lived in Canada, Australia, and the UK by the way.