When a water heater expansion tank is installed on a closed system with the inlet facing down, wouldn't it trap air on the water side of the diaphragm? Is there a way to bleed this air out or would that trapped air dissipate over time?
I believe that most brands allow installation in any orientation; however, some recommend an "air-purger" in the system to ensure an O2-free environment.
wouldn't it trap air on the water side of the diaphragm?
Yes, it seems that air could collect there. Air there would not affect function (it would just compress, along with the diaphragm, upon pressure increase due to thermal expansion). A disadvantage would be corrosion of the tank where air was present, but I would think that the diaphragm would fail (as they always do) long before the tank would fail due to corrosion. Another potential disadvantage is difficulty when replacing, as the water in the tank would empty onto you whilst unscrewing unless you drained the whole system (assuming no alternate way to drain).
There is an anecdotal advantage to mounting it this way: supposedly any debris/dirt would not collect on the diaphragm, supposedly lengthening diaphragm life...
Water heater expansion tanks come in various sizes. 2.1 gallon, 4.5 gallon, etc. The minimum size depends on your water pressure and the size of your water heater. You can do a calculation or use the vendor's sizing table. A water heater expansion tank bigger than the minimum is not a problem at all. A bigger water heater expansion tank is actually better in the sense that for a given water heater size, the pressure rise is lower.
What does it mean to have air trapped on the water side of the diaphragm? It means your expansion tank acts like a bigger one, until that trapped air dissolves. So you have nothing to worry about.