Learn Danish: Har han friet til dig endnu? - Has he proposed to you yet?

You are right that fri is commonly the adjective for free. However, here it is the past participle of the verb at fri, which means to propose marriage. The verb originally meant to woo or love, and it shares roots with the name of the Norse goddess Freja.

Why do we need the preposition til here instead of just saying friet dig?

In Danish, the verb at fri cannot take a direct object. You always use the preposition til to indicate who is being proposed to, making the fixed phrase at fri til nogen (to propose to someone).

Could I use allerede instead of endnu in this question?

You could, but it completely changes the nuance. Using endnu (yet) makes it a neutral question about whether the event has happened. If you swap it for allerede (already), it expresses surprise that he might have proposed so soon.

What is the simple past tense of at fri if I want to say he proposed yesterday?

The simple past tense is friede, so you would say han friede i går. In our example sentence, we see the past participle friet because it is paired with the auxiliary verb har to form the present perfect tense.