What does Damian think of Bruce? Does he respect him? Why did he want to be like Batman and not be like Ra's ghul or his mother? Maybe it's because he grew up without him and he liked how Talia or Ra's described Batman?

Yes he respects his father a lot. At first he didn't went to be like him. But bruce made him see that what he doesn't is good, what Talian and Ra's do is evil.

He respects him

Probably the person he respects most in the world or Dick Grayson, the two are at the top of Damian's list.

Injustice Damien probably hates his dad to cellular level, canon Damien does respect his father.

Damian definitely respects his father. In fact, he'd do everything to prove it to his father.

Damien respects him but always thinks he can do better than him.

He respects him--he's just a terrible kid.

There seems to be a low key Cold War between Damian and Bruce in Rebirth.

He loves Bruce but I'm starting to think there isn't much respect there these days. Bruce writers like King Snyder and Tynion ignore Damian while Damian writers such as Percy and Tomasi use that negligence to throw shots at Bruce. Others like Orlando and Williamson are also pitching in. See Button and yesterdays The Shadow/Batman.

The character getting whacked by this crossfire is Duke Thomas.