Sweet tea is quite regional (the South).
Iced tea's popular in much of the country, but I think making it at home has dropped off just like making many other things at home has (how many people make their own stock or broth anymore?). These days, a gallon of store-brand iced tea is just about the cheapest drink most grocery stores stock—barely more expensive than making it at home, really, so why bother?
Old-school restaurants, and especially diners, may have a mid-sized glass jar[0] of sugar with a metal lid and little metal flap on it (that flips up if you tip it, allowing some sugar out) left on the table, mainly used to sweeten coffee and iced tea. This is getting less common, in favor of little trays of sweetener packets, but you still see it sometimes. Used to be very common, 1990s and earlier.
True sweet tea (huge in the South, i.e. the part of the country that seceded during the Civil War, but much less ubiquitous outside it) is incredibly unhealthy. It contains more dissolved sugar than is easily achievable in room-temp-or-lower water/tea, by adding it during the boil, same as how you can super-concentrate sugar in water when making simple syrup (though it doesn't go that far) and it retains that sugar even when it's refrigerated. Sweetened iced tea has a much broader range of sugar levels, and may be relatively healthy, with only a little sugar, or very sugary.
Outside of the South, you'll often get sweetened iced tea if you ask for "sweet tea". It's fairly uncommon to see the real thing in Northern or Western restaurants. This sometimes leads to confusion, with non-Southerners going to the South and ordering sweet tea and getting something offensively sweet compared to what they're used to, or Southerners ordering it elsewhere and being disappointed with what they're served.
[0] Diner-syle sugar pourer: https://img1.etsystatic.com/040/0/6766475/il_570xN.635748831...
Note: this does not apply to bottled/canned teas, which are usually close to "half and half" in sweetness.
Most restaurants here serve soft drinks from a fountain rather than individual cans/bottles, so refills are free. If a restaurant uses individual containers for service, you will have to pay for refills. However, iced tea has free refills everywhere I've ever seen.
I forgot about iced tea, but I'm under the impression that how popular it is depends on the region of the US you're talking about. I don't see a lot of people drinking iced tea in my area, and I don't think I've ever seen anyone make it for themselves.