I'm sure ICE racing will continue in one form or another, but Formula 1 is enormously expensive; manufacturers justify it in terms of the prestige it brings them and the R&D work that can trickle-down into road cars. If the engine technology is completely different then both of those benefits become a lot more questionable.
Formula 1 chasses are way different to standard car chasses so presumably yes, but it really seems like an assumption to be careful about.
F1 chassis/aero/suspension innovations are more or less completely irrelevant outside F1. Switching from ICE's to EV's wouldn't change that.
Nah. Open-wheel racing has not been about actual research, already, for more than 20 years. The standard-bearer itself, Ferrari, was a racing team first and a manufacturer later; even today they are not in F1 for research but for marketing purposes (merchandising is basically their prime source of revenue). Open-wheel is largely a show, and it will continue to run as long as the show gets viewers, one way or the other.