Stupid question, is iced tea always with added sugar? That kind of would change the drink from moderately healthy (plain tea) to every doctor's nightmare.
No. You specifically order "sweet" or "unsweet" tea in the South (or "half and half" if you want sweet but not cloying). Anywhere else, I would expect that sweet tea would be unavailable.
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.
If you're in the South, Ordering "iced tea" with no qualifiers will probably get you sweet tea. Everywhere else, it'll probably be unsweetened. We do have a variety of less-sweet tea drinks, but those will be typically referred to by a brand name.
No. Sweet tea (obviously) always has added sugar, but iced tea may be sweetened or unsweetened. Restaurants that serve it may offer sweetened or unsweetened, or may offer only unsweetened (sometimes styled "unsweet"), with sugar packets to sweeten it yourself if you so choose. It's rarely (but not never—typically only when bottled, though, this is basically never an option at a restaurant that offers cups of the stuff) offered with artificial sweetener included, but some people add their own to unsweetened iced tea. Restaurants and drive-throughs often have both sugar and one or more artificial sweeteners available, in packets.
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.
