I had a quick look at the technical manual for the AirMini and it appears to be virtually identical to the AirSense 10 in features. I would setup the AirMini to be the same as what works on the AirSense 10. That said your 5-20 or 4-20 sounds like an out of the box setup and is not usually ideal for long term use. A wide range like that can be somewhat useful to establish what pressures you really need, but probably not an ideal setup for the best AHI results.
Having your maximum pressure set too high can lead to some discomfort, but really does not do much to the overall results. Setting the minimum too low can cause some comfort issues, and can also lead to excessive apnea events especially early during the night.
You may want to discuss the settings for your AirSense with your sleep technician to see if they could be improved. What I would look at is the Statistics Tab in SleepyHead, and specifically the 90% pressure. That pressure should be very close to what you could get away with if you only had a fixed pressure machine and not an auto adjust machine. It is also probably very close to what may be an ideal minimum pressure to use. And the maximum pressure could be as little as 1 cm above that. Say for example your 90% pressure average is 12 cm. You could set the minimum at 12 cm and the maximum at 13 cm. SleepyHead will tell you very quickly how good a job that is doing for you. If your machine keeps the pressure at minimum and your AHI is good, the minimum may be too high. If the machine keeps the machine at maximum and you are having obstructive apnea events with the pressure at maximum then it may be too low.
I look at the going to sleep ramp function as a pure comfort feature. It has little to do with your treatment effectiveness, but it can have a lot to do with how quickly you can go to sleep. What I like is to use the ResMed Auto Ramp. I then set the Ramp Start Pressure in the 7 cm to 8 cm range. I also set the EPR at 3 cm, but the Type as Ramp Only. Say you choose 7 cm, then what that would do is give you 7 cm on inhale and 4 cm on exhale until you go to sleep. Then the machine ramps up the pressure to your minimum (say 12 for example), and stops the EPR. At least that is what works well for me.
You can set up both the AirSense and AirMini with the controls on the machine. Here are some links to the manuals which show you how to set up each one.
AirMini Manual
AirSense 10 Manual
Edit: I'm not sure that SleepyHead supports the AirMini, and if so you can monitor what you need for pressures with the AirSense 10 machine, and then just use the same settings in the AirMini.