As Chancellor, Palpatine evaded detection for over a decade despite regular meetings with Jedi Masters not because of some vague, plot-convenient “shroud of the dark side” diminishing all Jedi’s perception, but because he cut himself off from the Force for limited stretches of time, thereby hiding his own dark side signature when in proximity to Jedi. That he would willingly leave himself vulnerable for extended periods is a testament to Palpatine’s well-deserved confidence in his skills of deceit and manipulation. (This may also explain why Palpatine’s face really became deformed in ROTS – it was a reaction to summoning so much Force power so abruptly, when normally he would reopen his Force connection more gradually.)
Why was Luke allowed to keep the last name “Skywalker” if he was supposed to be in hiding? Perhaps Anakin’s victory in the Boonta Eve podrace became something of a local legend, and so it became fairly common for human residents of Tatooine to adopt the name, meaning nobody would suspect Luke was actually a Skywalker.
The logic of the Mortis arc in ‘The Clone Wars’ falls apart if the Father’s death simply depowers the Son, effectively eliminating the threat he posed. But it can make sense if the Dagger of Mortis transfers the power of its victims to the wielder – if the Son had killed the Father with it, he would have become immeasurably stronger and been able to escape the planet. But when the Father used it on himself, he created a loop that destroyed his power entirely, thereby depowering the Son as well.
Darth Vader’s injuries on Mustafar didn’t actually inhibit his Force potential at all (other than maybe stopping him from generating Sith lightning). It’s purely psychological – his immense power is still but a fraction of what Anakin Skywalker would have eventually reached, for no other reason than that Vader sees himself as less than whole. (This one may actually have been confirmed or at least hinted at in official media, but I’m not aware of a specific source offhand.)