In order of paragraphs:
More than likely not, but it depends on how sensitive your eyes are. Some owners have been able to detect faint flickering on the 60Hz panels. Your mileage may vary, so be certain to buy from a place that will allow for you to return the set, if you are not satisified.
No, the jump in quality is, well, there is none. They are evenly matched sets. Where one exceeds, the other is serviceable, and vice versa. They end up being a wash at the end, with either model lines being great purchases. I much prefer the Vizio, however, because I am of the opinion that it is foolish to pay K-Series money for an edge-lit set in today's day and age.
You use the tablet for configuring the TV's settings. It does, however, connect via bluetooth to the TV, as well as wifi. The ChromeCast essentially works by the tablet telling the TV where to the get the content and then the TV does the rest. I do not know if this can be done via bluetooth. It's a damned good question and I would recommend calling Vizio's Customer Support line. They've been especially helpful every time I've called. 1-855-833-3221 in the US.
Whomever said that isn't wrong, but they're also not correct.
Check out my previous post. With HDR gaming, input lag is at 60ms right now, set to drop to 45ms at some point in the future. Is that within your realm of acceptable? Again, it all depends on your personal sensitivity. I couldn't notice the input lag in Deus Ex: Mankind Divided, but it is the only HDR game I can play right now.
I love my P65-C1, long story short. However, I am not you. I can only recommend the set, but, again, my best advice is to go for the P-Series. It seems to fit within your budget. Just buy from a place that you can return the set to if you are not happy.