Peri knows too much

I don't think Big Finish have, unless you count the long period of time that Five/Peri spent together with Erimem. Attack of the Cybermen addresses some of the post-regenerative side effects that the Doctor is still experiencing despite having believed that he was now fully stabilised in The Twin Dilemma. One of these issues has led Peri to become subject to a few lapses of memory, she's been called Tegan, Zoe, Susan, Jamie and even the Terrible Zodin, so it's not impossible that she asked him who exactly all these people were. Actually, given how flighty and prideful he was, he might have shown her who they were without prompting anyway.

Burning Heart puts it down to her having been shown some "holiday snaps" (most likely uncovered through some spring cleaning) and I think two or more of the short stories have her reminisce about the Doctor warning her of the dangers that lurk beyond the TARDIS. The funny thing is that Peri seems to know the Daleks only by reputation because when she sees one in Revelation of the Daleks, she runs straight towards it, believing it to be some kind of machinery. Timelash sticks out like a sore thumb when it comes to continuity references that seem to almost deliberately not make any sense (e.g. Jo Grant, two companions for the Third Doctor, Resurrection's time corridor, two Loch Ness Monsters, etc.). The Two Doctors at least matches up with what was discussed in Attack of the Cybermen.

References are funny things... There's a Sixth Doctor comic that mentions Trantor from Isaac Asimov's Foundation as a possible destination (but dismissed as too cosmopolitan by the Doctor) and another Eighth Doctor story that says he travelled to Terry Pratchett's Discworld (to my mind, with the late, great Sir Pratchett himself in tow). I reckon if you can mention the incident without the placement (mentioning that you have been shot at, but not when or where, for example), you fare a lot better in meshing with the greater whole.

Well... Four, really, if we're counting Mindwarp.

1. Our Peri who never left Thoros Beta.
2. The Peri who was set up as a warrior queen by King Yrcarnos and had grandchildren as seen in The Age of Chaos.
3. The Peri who returned home after one adventure and suffered badly in Peri and the Piscon Paradox.
4. The Peri who was set up as a warrior queen by King Yrcarnos and left to have further adventures with Sixy.
5. The Peri who became an embittered older woman who blamed the Doctor for abandoning her on Thoros Beta as seen in Bad Therapy.

Yeah... The Time Lords are usually a lot better at covering up their messes than that. Must've been that civil uprising.

Wasn't there also one who came back to Earth with Yrcanos who became a wrestler? Or did I just make that up?

It'd be clever of the Doctor to give new companions a crash course in the kinds of things he tends to get mixed up in and the more prominent enemies he's faced. At the very least it saves the viewing or listening audience an infodump of a monsters' history. After all, at the end of The Wheel in Space the Second Doctor shows Zoe Evil of the Daleks as an example of the kind of creatures out there in the universe, never mind it's a clever way to frame a repeat of Evil. It also makes a nice change from "who is the Master?" or "what are those pepper pot creatures?".

Clever idea, the Sixth Doctor has an unexplained desire to talk about Jamie with Peri, so when it comes to them actually encountering him and the Second Doctor there's minimal time wasted on explanations that would waste valuable screen time.

It would be wise of the Doctor to at least tell his companions about regeneration, in the event they end up in a multi-Doctor story, encounter other Time Lords or simply are still companions during a regeneration story and need to trust the new Doctor to get through the current crisis alive. It would also be useful for times like if the Third Doctor UNIT period is revisited or something like that.

Well they both know details when first encountering them on-screen.

Eegh, somehow that seems worse. Watching it for the first time in a theatre, I thought it was a bit of metatextuality poking through.

Yeah I thought it was clumsy metatextuality. Clara takes the role of the audience, who may be resistant to the idea of an older Doctor and need persuading to accept him by their old favourite. All with total disregard to the integrity of what character she had.

And Peri's knowing too much is the same. As the programmes audience became more insular and fan oriented, companions didn't need to query who the aliens were any more, because the audience already knew.

To the point that we had that very strange conversation between her and Vastra about the Doctor potentially being perceived as her "lover". Honestly, I think Moffat pulled that off much better for The Magician's Apprentice between Missy and Clara. I really, really enjoyed that later moment because it fed my own personal theory that Missy's madness is primarily an act for people she considers beneath her.

Exactly. It would have been much more interesting to show stories where good is as relative a concept as evil. In fact, I thought that was the road they were going down after Into the Dalek. Not so much, unfortunately. It all felt like things that had already been said and done, nothing really worth commenting on again. You don't even have to go back to the classic series to know that there are elements of evil inside the Doctor, the RTD era incarnations do demonstrate and are conscientiously aware of traits they themselves would consider undesirable.

Hell, Primeval deals with this problem with one line of dialogue from the Fifth Doctor:
In fact, these kinds of black-and-white moral judgments tend to come more from villains than they do protagonists in Doctor Who. The consuls in Primeval consider the Doctor evil because he dares to say that morality is a spectrum, the Doctor's acts in Trial of a Time Lord are deliberately thrown into a bad light by the Valeyard, the Daleks describe him as a terrorist in Blood of the Daleks, etc.