The real world reason is because Star Trek is an American TV series, and Picard has to specify that he wants the tea hot.
The thing is though that he's not just saying "Tea, hot," which would be a very vague statement. In my experience though, and this is speaking as an American who frequently drinks iced tea, Earl Grey is very rarely consumed cold. I don't believe I've ever encountered it chilled. Now, that doesn't mean you can't do it, but I can say right now that if I walked into a place that served both iced tea and hot tea and I ordered Earl Grey, it would come out hot.
As far as the iced or sweet tea issue, I would have to say from personal experience that, were I to walk into an average restaurant here in the south (SC resident), the immediate assumption on the part of the waiter, assuming it isn't a cafe that normally sells hot beverages, would be that when I say "tea", I'm referring to iced tea. In fact, while many may specify "iced tea" when ordering, I have seen many cases where people simply order "tea" and get what they want. A much bigger issue is whether or not it is sweetened, which is why you often hear "sweet tea" being used when ordering to differentiate it from unsweet. Whether or not it is iced is already a foregone conclusion. Now, I am very fond of hot teas and do often drink them at home. In my experience you simply don't encounter them often in most restaurants unless, as already mentioned, you're in some kind of cafe. In that setting you normally refer to the specific type of tea when ordering, of course, which brings me back around to my first point: "tea" can be interpreted as either iced or hot, but "Earl Grey" or "Chamomile", at least in the American Southeast, is normally always interpreted as hot. As long as you're operating under all these conventions of what "tea" means based on the situation, you're fine.
And, by the way, the reason why we assume tea is iced down here is pretty simple: it's hot. Really hot. It might not be as big of an issue in the present day, but before central heating and air conditioning, we were all much more subject to our environments. It seems to have left its mark in the types of food and drinks we prefer.
As for half-and-half, I've heard of that before but under a different name, although I can't remember what it was at the moment. I seem to remember hearing that it was popular in the SC lowcountry. From the name it kind of reminds me of my grandparents' tea (iced, of course). Although I don't think they actually used lemonade, they always used a large amount of lemon.