That was Vader’s plan all along, to have Luke kill the Emperor and then rule the galaxy as father and son.

Yoda was wrong. Maybe he needs to overcome his propensity for succumbing to fear to protect himself against the dark side but he also needs to defend himself.
The dark side makes more sense anyways. He should’ve joined palpatine, killed him, then subverted the dictatorship directly. Less blood shed, you have imperial resources to work with, everyone wins.
Stop.

He wasn't supposed to bring his weapons according to Yoda. It represents fear and anger. It never really amounted to anything, but Yoda/Jedi believe you can't do those things.

See, the problem with the failure in the cave is that Yoda goes against everything he says time and time again in the prequel trilogy vs. Empire Strikes Back and Return of the Jedi.

It's not only that he brought his weapons, a jedi is allowed to defend himself, but the important thing a lot of people won't notice is that he was the first to draw his sword and thus he is the attacker. Same thing again when he confronts Vader. He has no patience, hastes into battle and thus loses both times.

Luke was supposed to look "within" when confronted by his fear, and the object of his emotions. But instead, Luke went right for his weapon, struck out in anger and killed the spectre, only to then see its face as his own looking back at him. A vision that not only foreshadows Luke's parentage, but quotes that old Jedi proverb: "he who lives by the lightsaber shall perish by the lightsaber."

Yoda telling Luke that he didn't need weapons was strange, because JEDI use lightsabers all that time, and going into battle against the SITH without a weapon is stupid. I'm not sure what that was about
But, when I watch the fight scene, it looks like Luke is very angry, and probably filled with hate, because Vader "Killed" his Father, and he let those feelings control his actions, to win the fight
Yoda and Obi-Wan keep telling Luke that those things lead to the Dark Side, and that Luke should mind his feelings. Luke was also told he should be fighting with knowledge and skills, for Defense Not Attack.
When Vader's Mask showed Luke's face, I think that was to tell Luke that his actions were that of someone going to the Dark Side, and that he failed. Just like his father.

Don't be such a prude!
Your moralism is out of place in a galaxy far, far away (and long ago)

The fact that we know Leia is his sister makes this creepy and sick AF.

Great question. I bet he would have seen Leia in a sexy white frock with her hair down and a close-up of her pouty lips beckoning him to take her.
Right?

What would he have seen if he didn't take his weapons?

I think Luke's failure is that he saw Vader in the first place, and then that he saw himself in the mask. Whatever he saw revealed the fear in his mind

"Once more, Ackbar will rule the Galaxy, and… we shall have Peace."

His twin failures were reacting fearfully, and resorting to violence.
VOTE FOR PRESIDENT ACKBAR 2016!
" I assure you, my resolve has
never
been stronger!"

Luke's failure is that despite Yoda's insistence that "Your weapons. You will not need them.", he took them anyway, fulfilling Yoda's earlier statement (in response to Luke asking what was inside the cave) "Only what you take with you.".
Some see the Vader in the cave revealed to be Luke as a hint of Luke's parentage, but I see it more as Luke's true enemy is himself (his anger and fear).