Is my Trek FX2 really toast?
The fact that you've left the bike out in the rain the whole time and that the chain is rusty suggest that you've not been looking after it. It doesn't surprise me that it would now be damaged beyond economic repair.

I used to have a Trek 721, which I think is a predecessor model to the FX2.  Mine was kept under a roof and lasted longer than yours but, after several years of use, it too hit the point where enough stuff needed replacing that it wasn't worth doing so.  From the way it's been treated, it sounds plausible that your bike would now be at that point, too.

I suggest that you take it to another bike shop for a second opinion and then decide how to proceed.  Assuming that they say essentially the same thing, you have two options: either pay half the cost of a new bike and have a bike that will work for maybe another six to 18 months, or pay the whole cost of a new bike and get a new bike.