There are no rules. If there were, the rules would have been reported out by the Rules Committee and voted on by the House. Since there was no vote on Rules, and no vote on an inquiry, it is not the House that is proceeding with impeachment. It’s a group of democrat members, operating on their own, not as the House.
The House has the power of impeachment. Not the speaker, not a committee. That is why the House votes on an inquiry.