We middle-of-the-roaders are out here, don't worry. I guess the 2 extremes stand out more, that's all. I'm not sure that most people aren't in the middle - I think they are - but the better stories and greater drama are at the edges.
I think it's about balance, about choosing battles, and about attitude. For example, on the potty training - it's in the kid's control, not yours. For some boys, 3 is early for potty training. Your kid isn't ready. I get that you are sick of diapers, but what can you do? Let that one go. I promise he will not go to kindergarten in diapers, and he will figure this out. I believe in having a potty there, and in having books about it (you can always read them to your younger child and see if the older one listens in!). I believe there are techniques to use to train - but I also think they only work when the kid is developmentally ready. Mine was closer to 4 when he trained - it was incredibly frustrating but it was so much better when I realized that he had all the cards.
The food thing is a little more flexible. You can hide nutrition in food, you can figure out that he won't go hungry if he skips a meal just because he doesn't like what you put out, and you can keep offering things to try knowing that kids need something like 10 tries with a new food to know if they like it. You can take away his dinner if he doesn't eat it, and then put it back out in an hour when he's hungry - rather than making something new. You can put some sort of fun clock in his room and teach him what it needs to say before he can come out of his room. He doesn't have to sleep, but he can play quietly and he cannot wake up the house. Or put up room-darkening shades since the longer days brighten the room so early.
The 13 month old needs to put himself back to sleep. He needs the rest, and so do you. Is he waking up and wanting to nurse the way he does during the day? Or can he not self-soothe? It's long past time for that to get worked out, for everyone's sake. You may have the added problem of the little one waking up the big one, and then the big one is still up at 5 - you didn't say. But I can imagine the frustration if that's going on.
All 3 areas - sleep, food, potty - are the ones where kids exercise control and independence - we can't force them to eat, sleep or pee on the potty. We can force them into a car seat, control what toys are available, arrange their play dates (or not), and stick them in the bathtub. But those other 3? They've got us. Somehow, they figure this out.
I think you might want to choose ONE area for each child, and work on that. The youngest might need to be sleep-trained. He does not need to eat during the night, and he cannot use you as his lovey and comfort every time he becomes conscious. By the time you get this figured out, he will have mastered walking and won't be falling so much - LOL. Or, he'll learn to pick himself up and realize he's not broken! Once he can do that, he may ignore you for every pit stop - you can also try "in a minute, sweetie" instead of always being quite so available. I realize that screaming will ensue. But maybe he'll get bored with that more quickly than you think. Don't pick him up every time he falls, don't let him nurse when he's not really in need either physically or emotionally (choose the key times - you can read him well by now), and don't run into the room every time you hear a thud. Try redirecting him with a toy instead of endless comfort if he's not really hurt, just frustrated and using you out of habit.
For the older one, it might be time to work on staying in his room. If you can get that down in a week, then work on the food. For my son, who was a picky eater, I realized that I had helped create the situation by giving him his favorites and then getting him into a habit that was hard to break. There are many techniques on food (lots of books, lots of wisdom on this site if you ask for ideas), and lots of recipes for disguising new foods - I was a master at putting stuff in pancakes that you cannot even imagine. I turned it into a game for myself and kind of enjoyed it.
So I think it's important to choose the key areas to work on.
And yes, we DO understand!