@CHfather is absolutely right, listen to everything he says.
I can't stress enough how important oxygen is for the treatment of CH.
I can't say for sure you have CH, I am not a doctor. But I think CHFather is right that you should treat it like you do have CH and would strongly recommend you try out oxygen. I suffered with CH for 7 years before I got O2, and things completely turned around for me once I did.
Imitrex is a good starting point, but I can abort a headache with O2 with no side effects in about 10-15 minutes. Imitrex makes me feel tired and terrible and can make things worse if you take it every time you get a headache (every day).
My understanding of how oxygen works is that it has vasoconstrictive properties that cause the blood vessels pressing on the trigeminal nerve to constrict (get smaller) and stop pressing on the nerve. I am not a doctor, however, and this may not actually be the case with oxygen. Not a lot about how it works is actually known.
At any rate, some of your symptoms sound like what happens to me. The hard thing here is that nobody has the same symptoms and triggers. Like @jon019 said, we are all different, yet the same. 2pm is a normal headache time for me, I call them my "cluster times" (mine are 2pm, 6pm, 10pm, and 90 minutes after I go to sleep). I can practically set my watch to it. There are others that are triggered by any number or combination of things, and you are going to have to find out what yours are. Some of the medical literature says clusters don't have triggers, but that is not true. For example, alcohol is a trigger for a great number of us.
To make matters more complicated for you, you describe other symptoms that are not necessarily unique to CH. Some of what you describe we know here as "shadows," and there is almost nothing in medical literature about them, but most of us have them. There is a feeling in your head before you get hit with the intense pain, and it acts as a warning about what is coming. When the pain comes it is searing or hot and located behind your right eye. That pain starts migrating into the orbital area and reaches down into your sinuses and your jaw on the same side. That is because the trigeminal nerve attaches to all of those places, and is the nerve that causes our cluster pain. (CH is one of a group of conditions known as Trigeminal Autonomic Cephalagia)
Also included with the cluster attack is a profound tension in the neck. This created issues with my diagnosis of CH, because it is part of the mechanism in migraine as well. I have seen no medical literature on this, but anecdotally I have a few cluster buddies that talk of their "cluster knot" in their neck or shoulder while they are in cycle and complain of the almost painful tension in the neck and shoulder.
I also have sensitivity to light, especially where there is a large contrast (ex: oncoming car headlights at night).
As far as the sleeping it off goes, I can only do that with shadows after an attack. However, napping is a big time trigger for me, so I can't take a nap unless I want to get hit with a 10 later that day. When I did nap, I would wake up with that familiar feeling behind my eye. This is present for me any time I take a nap, even if I was feeling fine before I went to sleep.
Also, you need to start keeping a headache diary. This is important because if it is CH, chances are that they will suddenly stop one day, only to return later down the road. There are 2 types of CH, episodic and chronic, and most people (75%?) are episodic. Keeping track of your headaches, descriptions of the pain, what you ate that day, weather conditions, stressful events, etc. can help you determine what is going on. That is good information to have even if you end up not having CH.
I apologize for this being so long, I hope it is helpful.
I hope you don't have clusters. But if you do, you are absolutely in the right place. There are a lot of people here who understand and are happy to help you. Your first step is probably to get some O2 to try.
A question for the experts: If you can abort your headache with oxygen, doesn't that indicate that you do have CH and not one of the other TACs or migraine?