Even with Vader being damaged or being less than he once was, he was still facing someone completely untrained in lightsaber fighting and got his butt kicked. More to the point, what additional lightsaber training did Luke receive between ESB and ROTJ. Vader handled him easily in ESB....to the point he was essentially toying with Luke most of the fight. Isn't it only a year between ESB and ROTJ? You can't really bring up the Padme angle since she didn't exist at the time Lucas made ROTJ. Heck, she should have survived ROTS according to Leia's remembrance of her. Unless of course we are to believe she somehow recalls that she was pretty and sad for the all of three seconds that she saw her.
Luke didn't have training partners obviously, but he could have been training with remotes as we saw in ANH. I seriously doubt Vader was doing the same or getting any more training from the Emperor. Luke should be getting better, Vader should be getting worse.
Luke is entering his prime from an age and development standpoint. Vader is getting old and hindered by his armor. When you can't feel your own legs and arms and have had the same prosthetics for 20 years, I have to think your ability to sword fight is greatly reduced. Yes, a limited time since Bespin, but health rarely follows a static decline. Luke's got motivation on his side as well - what did Vader really care about by ROTJ?
The Padme angle is totally valid, or more directly the family angle. Luke doesn't install good in Vader, he helps bring the good back to the surface. And since we know there was still good in him, it's natural to assume that even a fairly bad person would still have reservations about killing his own son - practically the only family he has left. Pretty hard to channel your best when faced with the prospect of killing your boy. Anyone with a son should see that pretty clearly.
As for Leia remembering her mom, there's a ton of logical explanations there too. She's tuned into the force, so maybe she picked up her mom's sadness at birth and carried that forward. Did she see visions of her mom's life, similar to what Anakin saw of his mom? Seems pretty likely that Anakin's daughter could channel similar abilities, even if only for a few years. Do we even know that she's talking about her real mom? Heck, maybe she's even referring to Breha Organa, her adoptive mother.
It's all subject to your own interpretation I guess, but I don't find any faults with how either of these issues were addressed in the movie.
You are looking at this with a lot of revisionist history. You have to look at this as ROTJ in 1983 as a stand alone movie and what is shown on screen in order to make the comparison with Star Trek in terms of being illogical. The gripes that I have seen about Into Darkness mostly don't deal with previous Trek. As for Luke, sure Luke may
have been training with remote, but he also may
have been running so many missions for the Rebellion that he didn't have time to train. The only lightsaber training that is the top level of cannon is his initial training with a remote in ANH. Additionally, training with a remote isn't for dueling, it is for blaster deflection. He had one previous one on one duel and he lost badly.
As for Vader's health, in the OT there isn't any evidence that his health is in a rapid decline. A lot of the poor health of Vader stuff is from the EU. Until Luke beats him down, Vader appears to be fine.
True that Luke has more motivation, but it wasn't like Vader was on the defensive until he brought up Leia, which sent Luke into a rage. Luke's victory serves the purpose of the story, not the purposes of logic. It is hard to believe that Luke's first brush with the darkside would increase his focus since he has been trained to use the Force when he is calm and at peace. It would give him the extra power, but probably not the focus to wield it properly. Vader really only makes the decision to save Luke when the Emperor is frying Luke with lightning and Luke is begging for Vader's help. Who really knows what would happen if Vader turned Luke's anger against him and ended up disarming Luke. Maybe he kills him on Palpatine's command and secretly decides to turn Leia instead.
The reason I said Padme was invalid is because when ROTJ was made, Padme didn't exist. There wasn't a story written for her yet. As for Leia's memory, it is of Padme or the idea that would be Padme in the future. Luke specifically says "do you remember your mother, your real mother?". It is easy to speculate about the Force and use that is a possible way to have these memories, but it is a real stretch with what is shown on screen.