Learn Russian: В саду висит красивый гамак.

Because the hammock is already in position. The verb висеть (to hang) describes a state of rest (where?), which requires the Prepositional case. The verbs you are studying now, like садиться or ложиться, describe an action directed toward a destination (where to?), which requires the Accusative case.

I thought masculine nouns take an -е ending in the Prepositional case. Why is it в саду instead of в саде?

A small group of short, masculine nouns use an irregular -у ending when describing a physical location (often called the Locative case). You have likely seen this before with words like в лесу (in the forest), на полу (on the floor), or в шкафу (in the closet).

Why is the subject красивый гамак at the end of the sentence instead of the beginning?

Russian word order is flexible, and the most important or newest information usually goes at the end of the sentence. Starting with В саду sets the scene, and placing красивый гамак at the end emphasizes what is in the garden.

Does the verb висит have a matching verb for assuming a position, just like сидеть (to be sitting) pairs with садиться (to sit down)?