Learn German: Ich bin krank und mein Hals ist rot.

Yes! In German, der Hals is used for both the inside (the throat) and the outside (the neck). You have to use the context to know which one it is. Since we are talking about being ill (krank) and it being red (rot), "throat" makes the most sense here. You will also often see this in the word Halsschmerzen, which means a sore throat.

Why is it mein Hals and not meinen or meine?

Hals is a masculine noun (der Hals). In the second part of this sentence, it is the subject (the thing that is red), so it needs to be in the nominative case. The masculine nominative form of "my" is just mein.

Why don't the adjectives krank and rot have any endings?

In German, adjectives only get special endings when they are placed directly before a noun (like in the phrase ein roter Hals). When an adjective comes after a verb like bin or ist, it is just describing the subject and stays in its basic, dictionary form.

Does the word und change the word order in the second half of the sentence?