Wouldn't really need to be an issue if the other shows just didn't go out of their way to reference anything going on in The Flash's setting. Ultimately, these are all separate shows whose storylines need to stand on their own...and it's not likely to be a big deal to the majority of more casual viewers, just to the likes of us.In theory, no, but I think a lot of viewers would be confused if they didn't.
Then it's not about fixing the universe and saving everybody else, it's just about Barry getting himself back his home timeline.
Which still works...ultimately, it's about Barry's journey, and he's the main character of his show. Thinking further on potential complications in the storyline...Barry's not likely to just instantly come to the conclusion that he has to let his parents die to fix everything...he's liable to make at least one attempt to come up with a third timeline where everything's perfect, and of course it would just make things worse.
Also, if they wanted to pull some time travel illogic from LoT, they could say that the timeline that Barry altered still exists, but if Barry doesn't fix things soon enough from his own perspective, whatever new timeline he's caused will "settle".