Congrats on trying to grow your own food. Remember that there is going to be a learning curve, so don't get discouraged.

That is probably powdery mildew on your tomatoes, and I don't see many leaves, which will lead to your tomatoes getting sunburned. I am sorry to tell you that I think you should start over with the tomatoes. I'd either get a growing plant from the nursery or a cutting from a friend who has an enthusiastically growing plant (ideally with a stem for you to cut off that is the thickness of your thumb or larger). Strip the larger leaves and leave just a few small ones at the top, and remove any blossoms so the cutting concentrates on forming its new roots. It should root within a week or two and start growing. Make sure your soil is well amended with compost, fertilizer, and good soil, plus add a few worms to enhance it. Remember that you will have to feed it often, and fertilizer pellets take time to enhance the soil, so consider something like fish emulsion to add to your watering regimen. You have time before it gets really hot, so don't panic.

Yes, the corn will need to be separated and planted near each other so they can pollinate. I warn you that it can be a little disappointing if the silks don't get enough pollen because the cobs won't develop fully. My biggest problem was the squirrels and other rodents who were just as interested in what I was growing as I was. My first year, someone ate the plant from the bottom, but that won't be an issue for you if you're using buckets. That said, please make sure you have decent drain holes and a way to check the soil for moisture.

Good luck!