Hi and welcome John
It is impossible for us to diagnose if someone has CH as this takes a doctor with the appropriate skills and experience to do so, which normally means a headache specialist.
However it is very obvious that your wife has something wrong with her that is causing pretty severe headaches that needs to be resolved. Even if it isn't CH, it is something that is resulting in a significant impact on her.
From the symptoms there are multiple that are consistent with CH, but there are a huge number of different headache types where many of them also apply. Part of the diagnostic process is not just matching symptoms but ruling out all other possible causes (which makes it difficult to do).
One symptom confused me which was around Maxalt making it much worse. This isn't a common thing to happen with CH.
On the duration, single stabs are unlikely to be CH, but there are other headache types that can be of a similar short duration. It's also possible for someone to have multiple headache types, which again complicates diagnosis and treatment.
The standard diagnostic criteria for CH can be found at Multimedia File Viewing and Clickable Links are available for Registered Members only!! You need to

or

. Do note that the duration min / max values are not absolute, so a 4 hour headache would be uncommon, but it does not mean it can not be CH.
That site also has the definitions and diagnostic criteria for all the other headache types.
The link that John couldn't post is Multimedia File Viewing and Clickable Links are available for Registered Members only!! You need to

or

. From reading part of the site it is clear that she is no stranger to pain, so the fact that these headaches are much worse is a strong indicator that they need attention.
Whilst it is clear that the logistics of getting to see a doctor make the trip one that you don't make lightly, I'd strongly encourage you to do so.
I'd ensure that the doctor you'd see really does have the right skills and experience. I'd also prepare as much as you can for the appointment. I'd write down all you want to cover prior to the appointment, covering symptoms plus all the questions you want to ask. Do take notes during the appointment and tick off things so you know you haven't missed anything.
I really hope that you get this resolved soon as it is clear from what you have described that she is going through a lot at the moment. Do keep us updated on how you get on.