I doubt that a mask for using oxygen needs a prescription in the US but it is possible. Certainly here in NZ it doesn't.
The mask I use is from the web site store here - Multimedia File Viewing and Clickable Links are available for Registered Members only!! You need to

or

I've got some others that I got here in NZ (from BOC) which are not a patch on the quality of these masks and the 3 litre bag makes a huge difference.
Using a non-rebreather mask means that you'll be getting just the pure oxygen and not re-breathing the carbon dioxide that you're trying to flush out of your system.
For a load of great info about using oxygen effectively read the info at Multimedia File Viewing and Clickable Links are available for Registered Members only!! You need to

or

.
With questions, just ask away and people will try to answer them.
You're right about your newly minted GP not knowing too much about CHs. Even though a medical degree is several years only a few hours will be devoted to headaches with CHs probably getting a few minutes at the most as migraines and tension headaches will get most of the attention.
Breathing oxygen, apart from during a CH, will not make any difference and it certainly won't break a cycle. She may be getting confused with prednisone which for some people has broken a cycle, however for most people it just blocks CHs whilst a longer term preventive kicks in.
From reading posts from people the world over it seems that most doctors are reluctant to proscribe oxygen, partly from thinking it's dangerous (not with simple, sensible precautions) or it's not effective (it is very effective) or that it's expensive (not really and a LOT cheaper than imitrex).
Whilst 8 weeks may be fairly typical for many people, it varies considerably between people, so it shouldn't be used as an indication that they will definitely stop after 8 weeks.
I'd certainly rely a lot more on seeing an experienced neurologist who specialises in headaches. There are many, many headache types and other possible things which can all mimic CHs, so getting expert advice to get a correct diagnosis is essential.
Whilst 10 days may seem a long time, following the advice given by Joe with melatonin will help, plus using energy drinks like Red Bull (with both caffeine and taurine) will help to limit the intensity and duration of a CH.
The normal definition of an attack is a single CH, so for the example you used, you'll have 4-8 CHs (or attacks).
For shadows, these are probably low level CHs so they often have most of the symptoms of a CH but just don't quite seem to kick into being a full blown CH. Different people have slightly different experiences.
For other headaches, be careful about over use of other the counter (OTC) medication as using them too often can result in rebound headaches caused by using the medication too much. Work with your doctor on how to handle them too.
I hope this helps and keep asking questions...