We've established he's very strong willed- as in super duper 3 years old an very uninterested in not doing these things- normal since he's been allowed to form the habits. Based on your post it sounds like he doesn't get negative consequences for the actions. Not picking up the cups etc is just not reacting-aka ignoring-rather than teaching he is totally not allowed to do it.
Don't focus on each separate thing he's doing as a different challenge. Focus on teaching him to mind your warnings, and it will teach him the right actions.
Pick your firmest most effective consequence, as few words as possible, very brief at this age, and EXTREMELY unpleasant, so he chooses to avoid it at your warning. Use it every time for your second warning for whatever you're warning about. The consistency is key. If you use all different things at all different times, or things not very undesirable, he'll gamble, and it will take much longer for him to tire of the routine. Time outs will allow him to continue doing most of these things (especially whining and tantrumming) in the time out. Ignoring leaves it up to him how long he feels like acting like this. At 3? Already too long.
Make sure your home environment is very positive and loving, so his firm discipline stands out as a black and white lesson. If he's constantly getting punished (doesn't sound like he is) he'll be immune, and discipline won't matter to him.
If you make it about your warning, you can be more flexible. For instance, sometimes I let my kids scream in the house. If the weather was bad, they've been in all day, they just ate an energizing dinner, they feel like running wild and screaming? Fine. But as soon as I SAY no screaming in the house, they stop right away.
When he begins one of the behaviors, give him one chance to stop with a calm firm warning. When he decides to continue, zap! Calm firm consequence. He starts to whine-dont' let it escalate, one calm warning, "No whining." He continues to whine, ZAP! Repeat as necessary. Throwing stuff. One calm warning, "Do not throw food". He throws it anyway, ZAP! Repeat as necessary. Same for "No screaming in the house".
NO grudges, on with the postive day, he's only bringing the consequence if he chooses to, you're not angry. Don't ask a hundred times and get mad, or he'll learn not to begin to listen until you're getting mad. It boils down to enforcing what you say, and it will become natural to him to react to your warnings without discipline, and soon after he'll just want to do the right thing, not throw food etc. And he'll know whining is wrong, and when he starts, he'll quit when you remind him. My kids hate to hear kids whine. They forget they tried their best to start the habit.
It takes a lot of diligence at first-it's not for the lazy-and at 3, his habits are pretty deeply ingrained so you'll need consistency and it will get worse at first since he'll rebel against you taking charge, but it will click. You only have one week-you will not have it conquered by then at age 3, he's too mature and set in his ways, but you will have a good strong start. It will even become lighthearted and much easier over the next few weeks if YOU really do it.
We never allowed whining or tantrums, and at 2 1/2 my son never throws them, but the other day on a hot sticky day he had a cold and he wanted his juice, and it was after nap time, and I was engrossed in doing the bills and he started sort of whining for some juice. Immediately I said, "Hey, No whining." Usually that's the end of it, but he was so beside himself he laid down on the floor and did the "subdued whimper" so I said, "Oh, what??! Do I hear a customer trying to order a Smackeroni and Cheese?" and he hopped up laughing and went into the other room. Easy now, but it took diligence to get to that point. (and by the way, letting im whine and tantrum wouldn't be making him more creative and expressive, he is VERY expressive and communicates better without tantrums mucking up the water. The hubs and I were never allowed tantrums, and we're both very secure and expressive people)
Because he nipped it right away despite all the reasons to whine, I stopped the bills and went and got him the juice and a hug and set him up in his room all cozy. He knew the choice was black and white. Be firm!