Welcome to My Swiss Alps! It is really nice to get a post from someone who has clearly done their homework and though a lot about what they a hope to do.
I suspect that your post will generate a a fair number of answers.
I am a bit short on time to compose the thoughtful answers that all of your questions require, but here is a start. I suspect that others will jump in. I'l try top get back with a bit more in the next few days.
<<"Our flight out of Zurich is at 10:15 am. Would you recommend spending
the last night at an airport hotel, or would we be fine spending our
last night in Lucerne and taking a very early train to the airport?">>
I do that routinely ( Luzern to airport on the morning of departure- by the way, because the name you will see on signs around the German speaking region of Luzern is in the German spelling with a "z" , I use it out of habit) , but I prefer to be catching a plane a bit later. You'd be passing through the busy rail corridor from Luzern to Zürich during rush hour, and a few minutes delay (common near Walchwil and Zug) makes one nervous. If you do it, take the once per hour train that does not require a change at Zürich main station. That is the one leaving at 0710 ( or, xx10). if you do take the one at 0635, be prepared to understand exactly what your connection will be in Zürich main station. Use the timetable to learn what tracks the trains arrive and depart on, and take a look a the map of the terminal to get a feel for how complex the set of train platforms is both in and below the station.
www.sbb.ch/content/dam /infrastruktur/trafima ge/bahnhofplaene/plan-zuerich-hb-a4.pdf
One positive note - once you are at Zürich main station, if you do have to change trains there, and miss your connection, there are 5 to 10 trains per hour out to the airport. Some leave from "down under" while others leave on the main platform ( tracks 3-18). The signage is pretty clear, too.
A look at the map will help with that.
Personally, I'd stay in downtown Zürich, with your departure schedule.
Lots of good restaurants, and a nice place for a modest stroll before or after dinner.
Overall, Luzern offers more for the first time visitor, but Zürich has lot of cosmopolitan charms, and grows on you....slowly. Not worth a lot of time on the first trip; good for some time on each following trip as you learn the city.
<<"2. A friend suggested we should make time to visit Annecy, France. We
arrive at noon in Geneva and could travel to Annecy from the airport. Is
it worth pushing everything in the trip up one day?">>
What do you mean "is it worth it" ?
How can anyone advise you on the "worth" of a place, not knowing your values and preferences in any detail? Perhaps if your friend could explain why Annecy appeals ( it is a very nice place, of course) then you could compare it with the options at your Swiss destinations, using the links on this page below as well as other sources, and make a decision:
Personally, I think that the time is Switzerland is precious. Montreux has some nice options....mountainous, or on the lake. The variety available in Switzerland makes adding a day or more enticing. And, in the Jungfrau region in particular, weather is critical, changeable in May, and more likely to produce a sunny day for you if you have more time there. Taking away a day does not appeal to me. My opinion, for what it is worth.
Reading Number 3 makes me want to suggest that you travel approximately 3 weeks later. ;-)
The route from Zermatt to the Jungfrau region can be quite scenic if you avoid the tunnel between Visp and Spiez. For most, the best part of the Golden Pass route is the section up from Montreux to approximately Montbovon or Zweisimmen. Not sure if that would fit in, but I'll mention it.
Finally, in addition to the timetable linked above, this map is particularly useful:
If you zoom in far enough, you can see the rail lines, and the menus offer some useful information.
Perhaps you have already discovered it?