All valid and good points!
Well, what we found in our study group is that each of us found a different message! Probably due to our different backgrounds and experiences, and each one relevant to a message of preaching what the passage is trying to say.
Where you've both covered a particular area, I've not duplicated what you've put (so please don't think I'm disregarding what you've both said!)
I picked up on the beginning of Ch 11, "the time when kings go out to battle", David stayed at home and sent Joab instead. Not a crime, not a sin, but 'not what kings do'. If he'd gone to war 'as he should have done' ..., And then David got up "late one afternoon", implying he'd be idle, and in his idleness, gazed on a woman bathing. If he had looked away, what then...?
So, you could try thinking along the lines of a minor thing escalating, with David trying to cover things up, and failing, and committing multiple murders. I felt this is a lesson we could learn from, to stop sins before they happen or before they get "too big" to handle.
Someone else picked up about David's frailty, yet essential decentness, yet he is deemed worthy to be the ancestor of Jesus.
Someone else picked up how he shrugged off his child's death.
I've deliberately not said TOO much, so if you wish you can think a little more on it. There are other parallels and messages I think that can be drawn out.
From an historical point of view, it is believed that this story of David was written post-Exile, in a period which was generally regarded as anti-monarchic. If you are into Jewish history, you may like to think on that...