My UK passport still hasn't come in, though I've been charged for it. For some reason, the Home Office in London will not give me any status update on my passport application (I've got dual US/UK citizenship) so I don't even know if it's on its way! I just moved (within Seattle) so this is a stressful time for me.
Yeah, I don't think they send updates. I was lucky enough to apply in-person during a down time (i.e. fall, after the tourist season), and got my passport in 3 days, but they didn't say how long it would take (other than normally it's 6-8 weeks) or that they'd give me any progress updates, so I think the passport just shows up in the mail when it shows up.
My friend keeps saying things like "Oh [your cat] probably won't even make the trip" and "You shouldn't even bother taking her" all because she's 14. If the vet says she's healthy enough, this shouldn't be a problem, right? My cat seems nothing like a stereotypical 14-year-old cat. She's very agile and playful and I've seen really no change in her as she's aged. So, maybe I DON'T trust my friend to prepare her for travel if she doesn't see the point
Honestly, even if my cat only lives a year after I take her, it will be worth it to me.
Sounds reasonable to me. I just flew my 8yo cat out to NM (from NC) to live with family, and she handled the flight surprisingly well. (Flights, actually, she had a layover in Dallas.) She's had a bit of an adjustment period since getting there, esp. with regards to my relative's other cat, but overall has been in good shape. (The reason she went to NM instead of England with me is that she's pretty nervous and has chronic sniffles so I decided the desert would be a better climate than England for her.)
If I don't do that, she'll have to be on the plane when I go over. Is that better than some kind of delivery service? Again, I don't have much money. If anyone has used a delivery service, how much did it cost and how did it work?
I'm not sure what you mean by "on the plane", but just to clarify, this won't really be any cheaper. PETS scheme doesn't allow pets to fly in the cabin (which is the cheap option), they have to fly cargo (which is the expensive option).

So either way, she'd have to fly in cargo, it's just a question of whether you're on the same plane with her or not. (Some airlines I've contacted said there are extra fees if the pet is unaccompanied, some airlines have all-inclusive options, as well. So if your airline didn't include those fees, it would be slightly cheaper to be on the same plane.)
I haven't looked into the transport service, but my impression is that it's significantly more expensive than just doing it yourself would be.
Can you stress to your friend that it is YOUR cat and she doesn't get to decide what will happen to her? Once you've got the bloodwork done, it's just a question of getting the USDA to certify your paperwork and doing the final vet visit just before the flight, so at least if your friend did watch the kitty, there's not a whole lot for her to do.
Good luck!