A lot of the problems you described we are experiencing with DS aged 3.5. Here's how I attempt to deal with it...
Food: I just don't have unhealthy foods around or offer them. Occasionally at a party or somewhere else, he sees a plate of cookies and wants one and I say it's okay. But at home, what's available to eat, including snacks, is healthy. Some days he won't eat a single "meal" but he'll have some smoothie (milk, bananas, whey protein, berries), some crackers and peanut butter, a nectarine, some tamari roasted almonds, corn chips and cucumbers dipped in tahini-yogurt dip. All these I consider healthy (maybe not the corn chips) even though they're not real meals. I just don't expect him to sit down and eat, because if I do, I'll be disappointed. Eventually I imagine he'll want to eat real meals. Good food is something people really enjoy. He's not going to deprive himself of that experience forever. In the meantime, I just make sure what goes in his body has some nutritional value.
Sleep: my son takes a long time to fall asleep too. For example, last night he brushed teeth and used the potty at 7:00 and I didn't leave the bedroom until after 8:00. We read two books, listened to music, I drew animal pictures with my finger on his back, we snuggled. Finally, he went to sleep. I try not to draw too much attention to it. If he knows I want him to go to sleep right now so I can go do something else, he's going to stay awake. So I just relax and use it as a meditative time for myself. Luckily he likes music I like, so we listen to Niel Young at bedtime. Sometimes I try waking him up earlier if there's a prolonged period of time when he struggles at night with going to sleep. So I will sing him songs in the morning at 7:00 instead of letting him sleep until 8:00 as he might do on his own. Also, I try to make sure he gets plenty of activity during the day, and then after 5:00, we only do calm games, so he can start to wind down a bit.
Not listening: I ditto the last response. Often he hasn't even heard me. So I have to get down at his level and repeat myself. That works about 3/4 of the time. The rest of the time, he can be in a sort of wound-up place where he just can't listen. I'm not sure I have any answers for that. One thing I would try is, and it sometimes works, is getting involved in his play. Say, "I can see you're really involved in what you're doing. It's looks like fun. I'm going to play with you for a bit, then it will be time for you to listen to my idea." This might give the child a feeling of validation and connection to you, so that she can listen to you more easily. Also, I sometimes will distract him with something engaging to refocus his energy. "I have these beads here to make a necklace. I'm going to make one. Do you want to make one too?" I guess it depends on the circumstance. I often want him to calm down so he'll be in a better listening place. Sometimes I want to communicate a specific thing, like it's time to go. The former scenarios allow for more leeway.
Screaming: DS has been sort of intolerable lately. Really compatative and angry, screaming a lot too. This morning he screamed at me when I came into the bedroom to get him. He wanted papa to get him, but papa had already left for work. He continued to scream and complain at me for 5 or 10 minutes. I sort of ignored him and tried to be pleasant. "There's breakfast downstairs if you're hungry." Then finally, I squatted down to his level and said, "I'm not going to talk to you right now because you're screaming at me." (He had been trying to get my attention about something). He stopped screaming and within 2 minutes was eating breakfast and being totally pleasant. Sometimes I get stuck on the fact that, why do I tolerate this sort of behavior, it's so rude, I do so much for him, why does he treat me this way, but then I have to remind myself that he's just experimenting with feelings, making sure that I love him unconditionally, and while I don't have to be scared of his outbursts or tiptoe around him, I certainly want to give him space to express his feelings. This is on a good day. Sometimes I lose it and shout and slam doors. But that doesn't do any good. It just models a poor way of dealing with hard feelings.
I know what you're going through, and I know it's hard. I struggle with it all the time. My husband and I complain sometimes that we're in an abusive relationship: our son treats us like crap sometimes. But we're in it for the long-haul, right? So parenting techniques have to be aimed at creating trust, love, communication--not fear, blind compliance, and intimidation.