Adjective and Adverb?

The sentence structure is not identical at all. In the first sentence, which by the way is not standard English usage (we would usually say please reheat it, not please make it hot), the emphasis is directed toward the soup (a noun). Thus the modifying word must be an adjective.

In the second sentence, the emphasis is directed more toward the recommended action. Someone has previously written the sentence wrongly, and now someone must write it correctly. An adverb tells you how an action must be performed.

And in fact, if we say "the sentence is wrong," we are misspeaking by using an adjective. The "wrong sentence" would not be the one that was written wrongly, but simply not the one we wanted (as in "we took a wrong turn" or "we picked up the wrong suitcase"). In a sense, "the sentence is wrong" is shorthand for "the sentence is wrongly constructed." And writing a sentence is not parallel to reheating a soup.

1. It is used correctly. You can also say: Please warm it up or Please heat it up. Adjectives describe nouns - in this case you're describing a soup, you want it hot.
2. Adverbs answer questions such as "how", "in what way" - so the second sentence should have "correctly": How should I write it? Correctly. If you want to use correct you would need to use it to describe a noun in order for it to be an adjective. You can say ".., please write a correct sentence".

1. Yes, it's grammatically correct.

2. You would say 'correctly', because it is a verb; you're describing the action. 'Correct' would not make grammatical sense.

The difference between the two sentences is that in the first one, "hot" is not describing the action of "making", it's not an adverb, it's an adjective. In the second sentence, "correctly" describes the action of "writing", so it's an adverb.