Because Vader is his father

Good point. Plus, he was really badly hurt at this moment and let his guard down.

On the one hand, he's learned the awful truth that Vader is his father. But on the other, he's learned that his father is still alive and feels compassion for him.

I agree with this in a sense. Except I think it's more like Luke calling out to him to join the light side of the force. I think one can say they were both feeling each other's presence for the first time no matter what the intent.

More that Vader was preying on a vulnerable Luke who was battered, exhausted and had his defenses down. Obi-Wan famously said that "The Force can have a strong influence on the weak minded.". Clearly Luke isn't weak minded, but at that moment he doesn't have his guard up, which is why he responds with "Father" so earnestly. And by that point you have to think that Luke deep down believes what Vader is saying, otherwise he never would've reacted so strongly ("Noooooooo! beat Noooo).

"Force suggestion"? You mean like Vader made him say it or what?

I always saw it as Luke being in a
very
vulnerable state and Vader preying on that with a sort of "Force suggestion".

probably because he's yearned for his father all his life (or at least since he found out he was adopted)
"He's dusted, busted and disgusted, but he's ok"

It was sort of condescending act.
Luke answering Vader's call to him through the Force wasn't condescending. I don't think you are using the word you think you are using. There was nothing patronizing at all about that moment. For Luke, it was a moment of truth about betrayal, deceit, desperation, loss and pain. Vader wasn't even certain Luke had truly accepted the truth of the situation until their next meeting on Endor.

I very much believe that it was there for the audience, I have read on here of people saying that they themselves thought it was a ruse until RotJ however the scene is not breaking the 4th wall, meaning Luke the character wasnt asking for the audience but for himself

He believed it right away. The Yoda confirmation was added just in case kids didn't understand he was actually his father.

Because it was a fact, Vader was his father, there was no point denying a fact. Besides, he knew there was still good in him, that's why he directly surrended himself in ROTJ and brought him back
Yeah but he doesnt believe Vader when he first tells him and needs Yoda to confirm it in RotJ so even by then he wasnt sure

Because it was a fact, Vader was his father, there was no point denying a fact. Besides, he knew there was still good in him, that's why he directly surrended himself in ROTJ and brought him back.

I thought it normal of him to call him
father
. Actually, thinking about it it would be agains his character to ignore it, to ignore the truth. Deny would lead him to the Dark Side.

Yeah, but why did he choose to call him like that? Even if Luke searched his feelings, understood Vader was telling the truth, why he decided to admit it to Vader that he accepted him as father? It was sort of condescending act.

Do you mean other than searching his feelings and knowing it to be true and all?