I think you have a nice hypothesis, but a slight change of wording would be more correct. Boiled juice would have less active vitamin C but not less because it does not leave the juice when you cook it. Here is why.
Vitamin C or ascorbic acid is a type of molecule called an antioxidant. Antioxidant molecules (there are many different types vitamin E and many of the molecules that make berries red or blue) in the body are very useful because they prevent damage to cells, proteins and your DNA by reacting with other molecules that would oxidize (or take away electrons) from these important structures. When an antioxidant encounters an oxidant, the two react and the antioxidant gives up some of its electrons to the oxidant molecule. This reaction can change structure of the antioxidant molecule and effectively shutting it off to react with the next oxidant it runs into. This type of reaction is a redox reaction where one molecule is oxidized (the antioxidant or vitamin C) and the other is reduced (chemistry talk for gained electrons). Oxidants in the body can be metal ions like copper ions or free radicals. Free radicals can be created by UV light and breaks a bond in a molecule that leaves a free electron. Electrons like to stay in pairs, and that molecule with the unpaired electron looks for an electron it can to make a pair. If the electron it finds to make its pair is in your DNA it can cause damage to your DNA and possible lead to in the worst cases to cancer.

In general, a chemical reaction happens more quickly at higher temperature, since the molecules are moving around faster and are more likely to bump into one another and cause a reaction. Even in cold juice vitamin C can react with oxidants in the juice and be oxidized or "switched off" but far more reaction will take place at higher temperature. It would be a good experiment to see the temptress when the difference becomes noticeable.

Here's how you could test it: vitamin C is ascorbic acid. It makes the fluid it's in more acidic. Red cabbage juice is a pH indicator; it changes color depending on how acid a solution is. So, buy some red cabbage, extract juice out of it (by cooking it), and keep that juice aside. Then, subject your vitamin C treatments to a range of temperatures, including one that you did not cook at all. Finally, add some cabbage juice to each of your boiled/heated/not heated citrus juice treatments, and see which one(s) have different colors.

I think you have a nice hypothesis, but a slight change of wording would be more correct. Boiled juice would have less active vitamin C but not less because it does not leave the juice when you cook it. Here is why.
Vitamin C or ascorbic acid is a type of molecule called an antioxidant. Antioxidant molecules (there are many different types vitamin E and many of the molecules that make berries red or blue) in the body are very useful because they prevent damage to cells, proteins and your DNA by reacting with other molecules that would oxidize (or take away electrons) from these important structures. When an antioxidant encounters an oxidant, the two react and the antioxidant gives up some of its electrons to the oxidant molecule. This reaction can change structure of the antioxidant molecule and effectively shutting it off to react with the next oxidant it runs into. This type of reaction is a redox reaction where one molecule is oxidized (the antioxidant or vitamin C) and the other is reduced (chemistry talk for gained electrons). Oxidants in the body can be metal ions like copper ions or free radicals. Free radicals can be created by UV light and breaks a bond in a molecule that leaves a free electron. Electrons like to stay in pairs, and that molecule with the unpaired electron looks for an electron it can to make a pair. If the electron it finds to make its pair is in your DNA it can cause damage to your DNA and possible lead to in the worst cases to cancer.

In general, a chemical reaction happens more quickly at higher temperature, since the molecules are moving around faster and are more likely to bump into one another and cause a reaction. Even in cold juice vitamin C can react with oxidants in the juice and be oxidized or "switched off" but far more reaction will take place at higher temperature. It would be a good experiment to see the temptress when the difference becomes noticeable.

All of your questions about Vitamin C sound like good Science Fair projects. You can start at the highest temperature, and see how much Vitamin C is lost. Then you can test lower temperatures. The questions about sugar and freezing are also questions you can answer in your science fair project.

If you don't have enough Vitamin C in you, you get a disease called Scurvy. You can find the details on the internet, but I think your gums start to bleed a lot.

Different vitamins will break down different amounts when you heat them or boil them. That is because they have different chemical structures that are more or less damaged by heating.

Do you have a good way to measure Vitamin C for your science project?