Question: Does Kat ever wise up again to Kara's secret? That and why did she (Flockhart) leave the show only to come back?
Question: Why did they move the show to Vancouver?
Answer: For one, it's usually cheaper to film in Vancouver because of the tax breaks. But all the CW Arrowverse shows (with the exception of "Black Lighting") are/were filmed in Vancouver. So it makes it easier and cheaper for sharing set designs, crossovers, etc ("Black Lightning" filmed in Vancouver during its crossover). Originally "Supergirl" wasn't part of the Arrowverse when it was on CBS, but it was too expensive to produce and wasn't reaching the wider audience the CW shows had. So CBS moved it to its sister company, The CW.
Question: Why would Barry need Kara's help in gaining extra speed to return to his own dimension considering that the device he had on was capable of doing that?
Answer: It was an experimental device and was not meant to propel him into another world, it happened by accident. He need an extra boost of speed to get him back home.
Question: Why did they change her costume and hair style?
Answer: Another part of why the change was that it's cold in Canada in the winter and the old costume didn't protect her from it, whereas the new one is nice and warm.
Answer: I can't say for certain, so I'm surmising that the change was to "catch up" with the other Arrowverse shows, particularly Arrow and The Flash, whose lead characters have undergone various upgrades and redesigns of their superhero suits over the years; after four seasons, Kara's probably just felt a little stale. And I believe the change in hairstyle was Melissa Benoist's personal decision and had nothing to do with the character.
Answer: Yes, s02e22. As Kara leaves her office, she said "go get them, Supergirl" to herself. After season 1, the show switched from a CBS production to a CW production and filming moved from L.A. to Vancouver. Flockhart isn't really a fan of traveling to shoot, and didn't want to be away from her family for so long.
Bishop73