A few thoughts to clear up some of your confusion about treatment of central apnoea.
Firstly, elevated pressure does not always lead to an increase in central events. Sometimes it does but mostly it doesn't.
Secondly, not all central are caused by hyperventilation. In fact, most are not. When central events are the product of hyperventilation (breathing too much) that is when ASV is the appropriate treatment. It will only increase the amount of air it's pushing when the patient's own effort decreases, and when the patient's own breathing muscles are working it backs off. Thus it works to fill in the gaps in breathing and ventilate the patient as little as possible. This is desirable in cases of hyperventilation.
When there are central events and the patient is hypoventilated, which is more common in my experience, then BiPAP is the treatment of choice. It can fill in the gaps in breathing caused by central events and also increase the overall level of air breathed in and out by the patient. This is useful in raising oxygen saturation levels and decreasing CO2 levels. It is often used in managing conditions such as COPD, motor neuron disease, muscular dystrophy, severe scoliosis and more.
Both BiPAP and ASV have range of controls beyond just the 2 pressure levels which are set. They primarily affect the timing of breaths and can get rather complex. BiPAP in particular has a lot of different settings and different modes, which is why I so often say that it should be set up by technician with the appropriate training.
It's worth remembering that events which appear to be (and are therefore scored as) central events on a diagnostic study are not necessarily central. Also, it is indeed possible, although unusual, for central events to be treated by CPAP. So there is a very sensible and well-established protocol to try CPAP as a first step in treatment regardless of how the diagnostic study looks. This is the best way of being certain if they really are central events, and can also yield some data that is very useful in applying BiPAP to treat central apnoea. Of course, this is best done in a lab, where someone is watching and can swap to an appropriate Bi-level therapy if CPAP is ineffective and there are clear central events. It can also be done by trying CPAP at home, but this is much slower.
In your case I noted that there is an increase in you AHI when you are in REM. This is strongly indicative of obstructive apnoea and not central. I would suggest that your physician and technicians might know what they are doing and are following the appropriate protocol. The real test will be how they respond based on how well your treatment works.
That does indeed look like you barely qualify for mild OSA. However, if CPAP makes you feel better then why not? It is completely harmless. Perhaps an in lab study would reveal more.
Without doing specific research into that machine, I can tell you that the standard is for unclassified apnoeas to be included in AHI.
To get a clear picture of what's going on you really need a proper sleep study, although it is unlikely that 13 events in one night are causing that much disruption however they are spaced. Having said that, if CPAP makes you feel better then there's no reason not to use it because it's completely harmless. There are some people who's sleep apnoea is not significant according to all the guidelines, but it is still enough to make them feel really tired, and CPAP makes them feel much better. I'd just be thorough with your investigation before throwing too much money at it. Oximetry on its own is pretty limited.
They should absolutely let you try the mask you want. They can make suggestions but I don't think they should just ignore you. And there is no reason why you couldn't try a chin strap. It sounds like they're just being lazy.
I am a technician, and not a doctor, so I cannot give qualified medical advice. Also, the information you have given is limited but I'll do my best. If you are an otherwise medically normal health adult then that suggests that your oxygen levels are a little bit low but not alarmingly so. If you have a history of smoking, pulmonary fibrosis, or some other condition that affects your respiration then this would be roughly normal. It appears that your oxygen level dipped a little on a few occasions during the night (13 times for a total of 1.1 minutes). If this was measured over a night of sleep that would not be enough to cause concern in the normal course of events, and could easily be explained as entirely normal fluctuations in your breathing here and there.
If you do not have a history of smoking, or something else which would explain slightly reduced respiratory function, then it would be worth discussing this with a doctor because it is a little low (as far as I can tell from what little information there is). It's not panic stations, but it is worth investigation.
With a nose mask you should keep your mouth closed. Generally the best approach is to just relax and breathe as normally as possible. If you machine says there is no major leak, and if it's not waking you up, there is no reason for you to be worried about it.
It is risky to use BiPAP without it being set up by an experienced tech and with an in-lab study, especially if there is a significant difference between the IPAP and EPAP pressures. This can hyperventilate the patient and induce hypocapnia and Cheyne-Stokes respiration.
I have seen home studies both under- and overestimate AHI, in some cases massively so, but overestimation is far more common in my experience. Home studies use fewer sensors and therefore measure fewer things which would otherwise stop an event from being counted as an apnoea or hypopnoea. Also, as you mentioned, if they cannot tell when you are asleep or awake they will often count events that happen in wake which would otherwise be discounted, especially after sleep onset. Wake respiration is usually much more irregular than health sleep respiration.