well the cause for icing is generally temperatures at or below freezing (sometimes +2 but that's a different debate) and visible or present moisture. Don't have to be in the clouds to pick up ice, frost can and will form at night. In regards to the super cooled water, water can be super cooled from 0 all the way to -40 C, after that it becomes a suspended ice particle.

As for the climbing in icing conditions, generally speaking attempting to out climb the ice is a bad plan for light GA aircraft as it generally cannot be done. Depending on the severity of the ice, the performance of the aircraft and the altitude you are at, you'll never be able to climb above it. It can sometimes be done/be a good idea, especially if its a thin cloud layer or as you said there is a strong temperature inversion but generally speaking, no.

An additional factor to look at when it comes to ice, is what type of precip is occurring and what is the temperature. There are three types of ice, clear, rime and mix and they each form with different types of precip and temperatures.

When it comes to icing and GA aircraft the rule of thumb i use is that if it is +3 or colder at the MEA, its a no go. If there are pireps or airmets up for ice, also a no go.