ما لون القميص الذي تريدين، أحمر أم أزرق؟

In Modern Standard Arabic you often ask What is X? without an explicit هو / هي.
So ما لون القميص...؟ literally works as What (is) the color of the shirt...?
Adding هو (ما هو لون...؟) is possible but more emphatic/formal; it’s not required.

Because the definiteness here can be generic or context-based: the shirt (you want)—i.e., the shirt that’s relevant in this conversation. The following relative clause الذي تريدين also helps specify it.
If you wanted a more indefinite feel, you could also say something like ما لونُ قميصٍ تريدين؟ (What color of a shirt do you want?) but that slightly shifts the meaning.

الذي is a relative pronoun meaning that/which. It introduces the relative clause الذي تريدين = that you want.
It must match the noun it refers to: القميص is masculine singular, so the correct relative pronoun is الذي.
(التي would be for a feminine singular noun, like البلوزة التي....)

Because تريدين agrees with the person being addressed, not with القميص.
تريدين = you (feminine, singular) want.
If speaking to a man, you’d say تريد:
ما لون القميص الذي تريد، أحمر أم أزرق؟

Yes—Arabic often uses a verbless (nominal) question in the present.
Here, أحمر أم أزرق؟ functions like (Is it) red or blue? as the predicate completing the question about the color. The is is understood.

أم is typically used in either/or questions where you’re offering specific alternatives and expecting a choice between them: red or blue?
أو is more general and can sound like or maybe... / non-exclusive, and it’s also common in statements.
So for a direct choice question, أم is the classic, clean option.

A common fully vowelled version is:
ما لَوْنُ القَميصِ الَّذي تُريدينَ، أَحْمَرُ أَمْ أَزْرَقُ؟
In real writing, short vowels are usually omitted, and in speech case endings are often dropped.