Question re. Lhasa and AMS
 

I was looking into possibly visiting Lhasa, and I see nothing but horror stories regarding people getting altitude sickness there. Lhasa is around 12,000ft elevation. In the past, I have competed in athletics in Denver (5,280ft) and Copper Mountain, CO (~10kft elevation) with no difficulty. Given that I was able to play sports at 10k feet, should I be at all worried about visiting Lhasa for tourism?

No one can say for sure. It's one of those "You'll know when you get there" issues. People can chime in with their experiences here but they'll have no bearing on your ability to cope. I'm not even sure that previous experience is a surefire indicator. After multiple trips to La Paz (12,000) and Cusco (10,500), some have gone well for me and others have not. I don't think I've ever suffered from actual AMS, but I've had trips where the acclimatisation took a lot longer than others. I'd love to visit Tibet one day. The same concern has kept me from doing it.

Talk to your doctor about gauging your fitness for travel.

You will only read horror stories on the internet because these are the people who bother to express their experience. The majority who had no issue do not bother to go to the internet to say that it was fine. Same with airline reviews.
Best is to acclimatize with a couple of days in Xining.

I will definitely second the notion that there's no one-size-fits-all answer here, some people fare better than others. It also comes down to how fast you reach that altitude. The slower you take it the better you will fare.

Personally, I do well with altitude and wouldn't give it a second thought. (I live at a bit under 3,000'. Tomorrow I will be hiking over 10,000' with no acclimatization.) Her sister, though--I would worry about her in Lhasa even with taking days to get there.

I have never been to Tibet yet, but I advise the OP to buy some good sunscreen prior to the trip. When I was 16, I went to Snowbird/Alta at 9a, and ended up with a 3rd degree burn on my face, which required me to be wrapped up like a mummy for 4 weeks.

as noted in the past, have been totally fine with altitude. Never even thought about it before as a concern, but the horror stories I read when researching Lhasa have given me pause. How many days do people usually stay in Xining if acclimatizing there?

I've been. The most common issues are headache and sleep issues - they tend to pass in a day or two. Some people are more prone than others - e.g. many Asians have increased sensitivity.. I wouldn't worry about it.

Well then, how many days would you budget for Lhasa, what are some other must-see places in Tibet and how difficult is it to arrange permission to visit Tibet? Thanks!

You need to join a tour to visit Tibet. The tour agent will arrange for the Tibet permit.

Is it possible to do a tour for just me and my gf and avoid being with a group?

To my knowledge, this is definitely possible, but you still need to hire a tour guide (daily fee, plus their hotel/food).

Foreigners need special approval to go to Lhasa. The altitude reaction of Lhasa is acceptable to the general public.