OP, we have milder versions of these same issues periodically (not eating meals that I know are acceptable tastewise, gorging on $$ prepared snack foods that are not mealtime foods in this house, sneaking into the fridge and cupboards to take food items that they KNOW are not allowed as snacks, such a chocolate chips). My kids are three and five.
Here's what works for us:
1. Have eating guidelines that EVERYBODY respects. If kids are expected to eat oatmeal for breakfast or else fast until lunch, then any adults who are home at breakfast time should also be sitting down eating. If kids can't snack in the hour or two before dinner, neither can adults. If postdinner snacking is not allowed for kids, then it's not allowed for adults. You aren't trying to impose arbitrary rules for the kids
. You are trying to create a healthy food culture for all members of the household
2. Don't stock the objects of their current fixations. For example, in your shoes I would not be buying any cheese sticks, and I'd TELL THEM WHY - "You guys have been sneaking into the fridge and gorging on those things instead of eating your meals." Then buy them again in a month or so and see what happens.
3. Manners, manners, manners. My kids didn't eat things off the floor in restaurants, but they told both me and any other adult who served them, in detail, exactly what they didn't like about the food. That was a big struggle for a long time and SO INFURIATING. But on the plus side, my husband and I have developed some lovely table manners while modeling the correct social response to being served food.
A rallying cry around here is "nobody will force you to eat Item X. But you are not allowed to tell your dining companions that you don't like Item X. Talk about something else."
4. Others have mentioned this, but the free-eating zone is just a lifesaver for us. A basket of apples, a loaf of bread, a box of raisins, whatever works for you. For my kids, this has cut down HUGELY on the sneaking - it's not like I was REFUSING them apples when they asked for one out of the cupboard, but they prefer to get them on their own without consulting me, so I put a basket on a table that they can reach. As they are so young, I have to take it up a couple of hours before dinner if I want them to come to the table hungry (and that's huge for me, if a person is sitting down to the evening meal and they are NOT hungry, something is seriously messed up with how they ate that day).
Having seen all these bad habits - fixating on specific foods and eating multiple portions per day to the exclusion of other kinds of nourishment, in-depth criticism of meals served in a social context, gorging on processed foods outside of meal hours - going on amongst adults in my husband's family, I realize what the consequences are going to be if I don't teach my kids healthy food habits and instill in them good food manners. So I totally sympathize with you on this. As Americans, we don't have a workable set of healthy eating rules that are enforced by our entire culture. So you have to create a mini-culture inside your own home. It's a huge and ongoing task.