Read this enlightening article on why kids lie – I think it will help you gain some insight with your son. Though distressing for parents, lying is a sign of developing understanding about how social ilnteraction works. http://nymag.com/news/features/43893/
Your son's homework lie sounds pretty typical, in that he's trying to limit some of the scheduling in his life and open up some space for the natural things he longs to do. (And it is called "home"work, after all, so he may even logically assume that's where it should be done.) Play, including drawing, is still the primary platform for learning and self-expression for a 5 or 6yo. Most homework is not – especially in the younger years, it's busywork.
Homework for early elementary kids has never been shown to be effective at helping them learn, and in some cases turns them off and turns school/academics into pure drudgery. Then parents seek desperately for ways to "motivate" their kids. Motivation can't be manufactured for work that is meaningless or oppressive – schoolwork must have at least some element of discovery, accomplishment, and relevancy if kids are to remain interested in learning.
This is a shift from your original question, but it's important: I keep hearing from parents around the country that their young kids are being loaded down with homework. This is a terrible strategy for schools to employ, and I hope parents will educate themselves on this and advocate for their kids. Educator Alfie Kohn is a good starting point: www.alfiekohn.org/
As for the other lie, it may or may not be. There is a rather catchy book called Everybody Poops which some intrepid teacher might conceivably have used to capture the kids' attention. It is cute and educational, and has been used by many toilet-training parents who have a child who's afraid to poop on the potty. So I'd ask the teacher about that one before discounting your son entirely.
Keep being truthful around your son, and he will sort out what's true and why it's important. You could give him a quick lie occasionally when he asks you a question, and then point out that it's a lie, why it's distressing or misleading, and what the consequences would be to you if you actually tried to pass it off as the truth.