Learn Hardcore Dutch: De tafel waarop ik mijn boek leg staat in de woonkamer. - The table on which I place my book is in the living room.

Dutch offers two ways to form prepositional relative clauses:
1) Preposition + waarop at the front: “de tafel waarop ik mijn boek leg…”
2) Relative pronoun die + preposition at the end: “de tafel die ik mijn boek op leg…”
Version 1 (with waarop) is more formal and keeps the preposition together with the pronoun. Version 2 is very common in speech. Both are grammatically correct.

In the main sentence, why does staat come after “De tafel waarop ik mijn boek leg” rather than immediately after De tafel?

Because of Dutch V2 (verb-second) word order. The entire phrase “De tafel waarop ik mijn boek leg” functions as one constituent (the subject) and occupies the first position. The finite verb staat must then occupy the second position in the main clause:
1st element: De tafel waarop ik mijn boek leg
2nd element: staat
remainder: in de woonkamer

Within the relative clause “waarop ik mijn boek leg,” why isn't leg moved to the very end as in a typical subordinate clause?

Relative clauses introduced by words like waarop, waarin, waarmee, etc., behave like V2 clauses. Here waarop is slot 1, ik is slot 2, so the verb leg follows in slot 3. That’s why it isn’t shifted all the way to the end.

Why is the verb staan used for de tafel instead of liggen?

In Dutch, objects that are upright or “standing” use staan, whereas objects lying down use liggen. A table is an upright piece of furniture, so you say: de tafel staat ergens.

Is a comma needed before or after the relative clause?

No. In Dutch, restrictive relative clauses (the ones that define precisely which table you mean) are not set off by commas. You write it continuously:
De tafel waarop ik mijn boek leg staat in de woonkamer.

Can I rephrase the sentence as “De tafel waar ik mijn boek op leg staat in de woonkamer”?

Yes. That’s a very common spoken alternative. It separates the preposition op to the end of the clause:
De tafel waar ik mijn boek op leg staat in de woonkamer.

Why do we say mijn boek instead of het boek?

mijn is the possessive pronoun my, indicating ownership (my book). het boek would just be the book. Use mijn when you want to show that the book belongs to you.

Could I omit the article de and just say “Tafel waarop ik mijn boek leg…”?

No. Dutch generally requires a definite or indefinite article before a noun phrase. You need de tafel (the table) or een tafel (a table).

Why is the preposition in used for de woonkamer?

Dutch uses in for enclosed spaces (rooms, boxes, etc.). So to say “in the living room,” you use in de woonkamer. Using op or another preposition would change the meaning or be incorrect.