Learn Dutch: We hebben het eerste document door te nemen. - We have the first document to go through.

This structure expresses an active task or an obligation. Just like in English when you say 'We have a document to read', using hebben (to have) along with te and an infinitive verb shows that going through this document is a task assigned to you.

Why is te placed right in the middle of door and nemen?

The verb being used is doornemen (to go through), which is a separable verb. Whenever you need to put te before a separable verb, it gets sandwiched exactly between the prefix (door) and the base verb (nemen), written as three separate words: door te nemen.

Could I just use moeten instead of hebben te here?

Yes, you could perfectly say We moeten het eerste document doornemen (We have to go through the first document). Using hebben te means almost the same thing, but it sounds slightly more formal and emphasizes that the document is a specific task currently on your plate.

Why does eerste end with an -e?