These are really good questions...and it's important to take time to think about and research them. There are a lot of bold claims out there on the web, including some misinformation. I'll do the best to answer these on the basis of what I know.
1. Black, green, and white teas do not have the same amount of antioxidants...and in fact, within each of these categories of tea, individual teas also vary widely in the amount of antioxidants. I read an interesting study that studied a large number of individual teas and measured the content of catechins and flavonoids (both types of antioxidants) and found that the antioxidant content varies hugely from one tea to the next. It's a bit technical but if you want to read it it's here:
The chemicals in tea that are beneficial to your health are the same aromatic chemicals that give each tea a unique aroma. So...if teas smell different, they have a different chemical makeup. And in general, the chemical composition of each tea is unique! Tea is so complex!
2. There are many more differences between black, green, and white teas beyond the length of time they are oxidized/fermented. Also, green tea does have caffeine. Some websites, including the USDA and other government websites, will claim that green tea has less caffeine than black tea. This is not true! The caffeine content varies widely from one individual tea to the next--depending on how it is processed, what size of leaves were used (and how many buds/tips--buds contain more caffeine), what time of year it was picked, and then, when you brew it, how much leaf you use (or how much leaf is in the teabag) and how long you let it steep. More leaf + longer steeping = more caffeine.
Also, black tea can contain more caffeine because the process of making black tea breaks down other chemicals in the tea, but doesn't break down the caffeine as much, so more caffeine is left over in the final tea. Some teas, however, like Pu-erh tea, have been aged so long that the caffeine actually starts to break down, so they can contain less caffeine.
3. Iced tea and hot tea contain the same chemicals, unless you brew them differently. If you brew tea leaves cold or at room-temperature (like sun-tea), you'll get a different chemical makeup from if you brew them hot and then cool it down. Sometimes there's a big difference even from using boiling water vs. using 180F degree water (slightly below boiling, about when small bubbles first start to form). Green tea is generally supposed to be brewed with 180 or so degree water, not boiling water...if you use boiling water with some green teas they can become acidic, making them not only taste bad, but be rough on your stomach as well.
4. Yes and no. The exact chemicals in tea are different from in red wine...but in terms of classes of chemicals, there is some overlap. Tea, red wine, and fresh fruits and vegetables all contain flavonoids, a class of antioxidants. On the other hand, most of the focus on antioxidants in tea are on Catechins, chemicals that are not unique to tea but that seem to be most highly concentrated in tea.
I hope this helps! I have a little bit of information on my website on these things...you may want visit my pages tea and varieties of tea. This page links to pages about black, green, and white tea with a little more info on what each of these varities is. Also, you might want to look at oolong tea...it's a semi-oxidized tea (between green and black) that also has health benefits.