well I've been using FCP7 for some time on OSX 10.10, Yosemite.
it working for me, but with some minor bugs,
the most annoying being that the colour picker doesn't work as it used to, and colour matching now has to be done manually.
if you don't do a lot of colour grading it won't bother you.
gradually it'll work less and less well.
what's the minimum OS you need for Chrome to work the way you want it to?
on another note: FCP 7.0?
that's pretty buggy in itself.
i'd advise you to do a software update and take to to 7.03.
nick