Caitlyn doesn't know Barry went past Mach 2 as only Barry remembers that he time warped.
As to everyone should have been instant erased and rewritten immediately, no. There is the ripple effect that can occur when time is changed. Things appear to be normal but then the ripples catch up and things change, or a temporal storm such as the one they are witnessing occurs and THAT is the source of erasure and change.
The 90's X-men cartoon had a two parter where Bishop have to save history by preventing the death of young Xavier. When he went back to get help from the Xmen the time change had caught up to the xmen and their world had changed but the time ripples/changes had not yet reached the year Bishop came from.
Doctor Who: it took all 13 Doctors to do this, but they all teamed up and slightly revised the end of the Time War to save Gallifrey. At the end it turns out that due to the ripple effect and time being out of sycn that only Doc 11 and above would remember and that War Doc and Doc 10 and 9 would think that Gallifrey was nuked.
The immediate effect of RF being erased makes sense as he and Eddie are the focal point of the change. Everything else is the ripple effect.
As to saving his mom, many things would change but would it be for the worse? hard to say. However attempting to time warp to save his mother at the risk of everything is rather selfish to say the least. The noble thing to do was let history take its course. Still stinks though.
I think the multiverse already exists as shown by Jay's helmet appearing. If Rip HUnter doesn't arrive to stop all this from occuring by going back about 10 minutes and prevent Eddie from shooting himself and offering RF a trip home, then I can see Barry stopping the singularity but getting tossed onto Earth 2 for a time.
Once RF is restored, the recording if ti was made will be there. henry could be freed.
If I was Eddie I would have gotten a (snip,snip) vasectomy instead of shooting myself O_o