Sure, it's alright if he wants to charge you for work he does around the house. Of course, following his logic, he won't mind paying you for what you do around the house for him! Last I checked "rent" included the roof over your head and nothing more. So logically anything beyond that is a "service" for which you should be paid - cleaning, cooking, laundry, etc. Of course errands will also incur a charge - picking up his groceries for example. And don't forget to include any overtime and an extra charge for "emergency" requests. Oh, and does he have a car? I wonder what the going rate is for a parking spot in your area. When I lived in a rental place, the laundry cost $1.75 per load (and another $1.75 for the dryer) - and that's if he does it himself. There's an extra charge, of course, if you do it for him.
In my opinion, he's being a jerk. Even if he wasn't living with you, you'd think he'd help you out with the minor carpentry work! The fact that he lives there at what seems to be a discount rent and won't do it is really low.
Next time he tries to charge you for his time, etc. tell him that's fine. Then tell him his rent's gone up to reflect increased home maintenance/improvement costs!