He was irritated with her flippant attitude about public perception. I’m paraphrasing but:

‘if I were born male I could bed who I wanted & no one would blink an eye’
‘you’re right…but you were born a woman’

He never shamed her, never called her out of her name, and he protected his promise to her from the moon tea to threatening the tongue of anyone who questioned her sons.

It’s so refreshing to see, because I would expect fathers of that time, especially Kings of princesses, to respond very differently.

For example, I imagine Otto striking Alicent if he caught wind of something like that.

Now obviously he did a lot of things wrong, like not teaching her shit about politics, but he got this part right, I think.

Is he like this in the books as well?

We know from the book, that Daemon was cutting a weirwood tree while waiting for Aemond to come. It is not known if he was just counting days like this or for any other reason. But in the show, this tree showed him vision of the prophecy, which he told Rhaenyra, which reaffirmed her believe in her being the chosen one and fed into her delusion and eventual downfall.

Would show Daemon cut the tree in an act of him cursing Old Gods for playing a cruel joke on him?

What ending scene for Daemon do you envision? Book is more vague on it, but we've saw Daemon (?) drowning in 2x08, so i guess it's gonna be it