Although all single wall rims and all double wall rims are not created equal, the rule of thumb is that indeed the double wall rims are stronger than the single wall.
Rim width should be measured by the gap between the sides, not externally. Your 22mm rim should work fine with the 54mm tyre, with only a slight risk of the tyre rolling off the rim if/when you get a sudden/fast puncture or if running abnormally low pressures. That slight risk is reduced by the stiff wire bead and stiff wall of the Schwalbe tyres.
The (very) conservative rule of thumb from road cycling is that a rim will suit a tyre of 1.4 to 2.0 times it's width, but especially in larger sizes like mountain bike tyres that
conservative limit can easily be taken as 1.4 to 2.5 times.
"Conservative" means that the tyre will not roll off the rim if you get a puncture, or slip off at the bead during a turn, will not pinch flat easily or risk bashing the rim on stones, and will have the proper cross sectional shape intended by the manufacturer. "Limit" does not mean you have to stay within those "limits", just that you should think about whether you have any special reason for going outside them, and don't combine outlandish sizing with outlandish pressures.
In short, a 1.6" to 1.85" tyre would be a great match for your new rim, but your 2.15" shouldn't be a problem at conservative tyre pressures.