Are you allowed to submit a letter to E6 promotion board without being on adverse action?

The E6 Board is local. You will be in the room, and the first thing they say is "tell us about yourself". If those 1SGs really need to know something, there's your chance. The senior NCO boards are held remotely, so letters are more prevalent and essential.

Sure, but unless you are really in need, or have a damn good understanding of the Army Writing Style, it typically does more harm than good. You're not going to impress a board if you can't spell, your grammar is horrible, and your letter flows like you forgot your ADHD medication.

The cited AR does allow you to write the board, but you can only provide certain information. It talks about pointing something out in your record. Make sure it's record information that the board actually sees. Pointing out stuff the board isn't allowed to look at demonstrates your ignorance. Most important is to make sure the stuff that actually gets looked at is accurate. There's an easily Googled web page that talks about what to pay attention to. My board experience is obviously with a different service and at this time rather dated. I don't know if the board staffs screen out letters that don't comply with guidance. One thing for sure, take a look at prior board precepts and get a drift on the points of emphasis the convening authority wants out of that board. They change over time but typically not much from the most recent year.
https://www.hrc.army.mil/content/REVIEWING%20BOARD%20INFORMATION