That changed between seasons 1 & 2. In season 1, Rittenhouse was a secret conspiracy that already controlled the United States and had been shaping its history behind the scenes for centuries. Garcia Flynn stole the Mothership in order to go back in time and destroy the organization, which incidentally meant pretty much destroying the United States as a side effect, so the heroes had to chase Flynn back in time and stop him from changing key events in American history. So Rittenhouse in season 1 didn't want to change history, but to preserve it, and they were essentially running Mason and his team from behind the scenes (although they did implicitly want to control time travel to give themselves more power). Eventually the team learned about Rittenhouse, broke from its control, and found a way to bring it down in the present without undoing the past, although it was revealed at the end of the season that there was a deeper facet to the organization than the one they'd brought down.
Season 2 changed the dynamic so that Rittenhouse was no longer the entrenched power trying to preserve a history that favored it, but a subversive group trying to change history in its favor -- mostly by undoing generations of progress in civil rights and putting women and non-whites back into servitude. It was a much more straightforward us-vs.-them dynamic than the first season, a lot more simplistic, and it didn't mesh very well continuity-wise.
No, it's consistent with the way time-travel fiction usually works -- the people who traveled into the past are from the original timeline. They left it, made a change in the past, and then came back to a different timeline than the one they left, so they still remember their native timeline while the people who stayed in the present were changed along with it. This is standard. When Marty McFly came back to 1985, he still remembered the original timeline where his parents were losers and Doc was shot, and he was surprised by the changes. When Kirk and Spock came back from the past in "Yesteryear," they still remembered the timeline where Spock was the first officer, even though nobody else remembered Spock. In the Arrowverse, when Barry Allen changed the past in "Flashpoint," he still remembered what the original timeline had been like (e.g. Diggle having a daughter instead of a son) even though everyone else knew only the altered timeline. It's time-travel fiction 101: The time travelers themselves still have their original memories, because they aren't native to the new timeline they created. (Although some stories do have the travelers' memories change. Legends of Tomorrow is deeply inconsistent about this -- sometimes the Legends remember the way time is "supposed" to be, but sometimes their memories are altered, depending on the needs of the story.)
For all of Timeless's sloppy temporal physics, this is one thing they did always keep consistent: The time travelers remember the timeline they left, while everyone who stayed in the present is changed by the time travelers' actions in the past. So every time the characters came back to the present, they needed Mason and Christopher (and Google) to tell them what history now said, and the others needed the time travelers to fill them in on how history had been before. In this case, Lucy and Wyatt were still in the 19th century when Flynn went forward to 2014 to re-kill Jessica. So they weren't affected by the change to post-2014 history.
The point is, you can't have it both ways. A show that's built around constantly changing and rewriting history cannot suddenly claim that history is fixed and consistent; it's a logical contradiction.
No, that's not the way this kind of time-travel story works. The whole point is to change things so that the original circumstances are erased from the timeline. Kirk and Spock prevent Edith Keeler from being saved. The second Terminator prevents Skynet from ever being created (ignore the later movies). Marty prevents Biff from using the sports almanac to get rich. Once the disruption to time is undone, you don't have to go back and make it happen again. Because the time travelers are from the original history and still remember the event even though nobody else does. So the catalyst for their change to history still happened for them, but it's been erased for everyone else and there's no need to recapitulate it. A loop and a branching-history model are mutually contradictory.
