No they cannot. They cannot open cabinets or doors including the fridge or any drawers that is an illegal invasion of your privacy if they do. Testing for lead paint does not require being nosy and going through your things.
I would, too, be very upset about them opening drawers, doors, and the fridge and taking photos but unless they attempt causing trouble with it I wouldn't cause trouble with them.
They can inspect at any time, as often as they want, as long as they give 24 hours notice and get permission to enter. You don't have to give them permission if they're being unreasonable with the time they want to do it, how often they want to enter, the reason(s) they want to "inspect", etc. They cannot invade your privacy.
If you're not doing anything wrong (not having a perfectly clean fridge is NOT a crime or anything wrong)...I also suggest not making a stink about it unless they're being unreasonable, an annoyance or inconveniencing you.