Learn Dutch: Ik ben te moe om in de rij te wachten. - I am too tired to wait in the queue.

They have two completely different functions here! The first te means "too" (as in te moe, "too tired"). The second te goes right before the verb to form the "to" in "to wait" (te wachten).

Could I drop the word om and just say Ik ben te moe in de rij te wachten?

No, you need om here. When you use te + adjective (like te moe) to show that a situation is too extreme for an action to happen, Dutch requires the om ... te + infinitive structure for the action clause.

Why does in de rij go between om and te wachten?

In an om ... te clause, om always sits at the very beginning of the clause, and te + infinitive (te wachten) gets pushed to the very end. Any extra details, like where you are waiting (in de rij), are sandwiched in the middle.

Does rij mean "queue" or "row"?

It means both! While een rij literally translates to "a row", the phrase in de rij wachten (or in de rij staan) is the standard Dutch way to say you are waiting "in line" or "in a queue".