Learn Russian: У меня в руке горячий напиток.

We generally drop есть when the focus is on where an object is or describing its details, rather than just stating that it exists. Since the sentence focuses on the location (в руке), the word есть is naturally omitted.

I previously learned that "hot" is жарко. Why does this sentence use горячий?

Жарко is used for environmental heat or how someone feels (like летом жарко for "it's hot in summer"). For physical objects that are hot to the touch, such as food or a напиток (beverage), you always use the adjective горячий.

Doesn't the word рука mean "arm" rather than "hand"?

In Russian, рука refers to the entire limb from the shoulder all the way down to the fingertips. It translates to either "arm" or "hand" in English depending entirely on the context.

Why is the ending changed to в руке instead of keeping the dictionary form рука?

The phrase describes a stationary location (where the drink is), so it requires the Prepositional case. For a feminine noun like рука, the -а ending changes to -е when following the preposition в.