how to clean cloudy antique glass jug?

Warm water and a mild detergent work best for routine cleaning. Avoid putting old glass in a dishwasher, as the heat and harsh chemicals cause permanent clouding. For stubborn calcium deposits, a soak in white vinegar often lifts the haze. Swill it around for a few hours before rinsing well.

My grandmother left me a heavy jug that she always used for lemonade in the garden but I was wondering if it was actually meant for a different drink originally?

Heavy jugs from the nineteenth century were often intended for water or ale. The thick walls helped keep the contents cool during meals. Look at the shape and any engraving. Narrow necks often suit wine or claret, while wide-mouthed, barrel-shaped pieces point towards ale.

what makes a glass jug heavy?

The weight usually comes from a high lead oxide content. English glassmakers added lead to the mix from the late seventeenth century onwards. This gave the finished piece a noticeable heft.

I have noticed some jugs have very rough patches on the bottom that can almost scratch a table top, so why weren't they finished off smoothly by the maker?

That rough patch is the pontil mark. It happens when the glassblower breaks the finished vessel off the solid iron rod used to hold it during shaping. Early makers left these scars exactly as they broke. By the nineteenth century, most workshops ground the marks down and polished them smooth. Leaving it rough was normal practice for earlier, everyday domestic wares.