Transistor-Switch Basics
1) Power dissipation in the active region usually is much higher than in saturation. So most of the power is dissipated during switching time rather than in conduction (saturation) as you can see in the (last) graph below:



(Source Power Electronics Handbook)

This puzzle is best explained by looking at a (simplified) model of a [power] transistor (which ignores things like quasi-saturation) using a so-called (DC) load line:


in cutoff the transistor has high Vce (determined by the external circuit) but zero collector current. So power dissipation is zero.
in the active region both collector current and Vce can reach high values (moving toward the upper left on the line below during turn-on) so their product (=power) also reaches high values.
once saturation is reached, collector current still increases (a bit), but there's now a dramatic dropoff in Vce, so power dissipation falls off dramatically as well. In the more sophisticated model of a power transistor, this droppoff is not as sudden because there's a quasi-saturation region (not shown in the picture below).




2) You're confusing the voltage spec of the solenoid with the Vce of the transistor. At 1.1A you only have about 0.7V Vce in saturation for TIP120; see operating point (red X) I marked on the datasheet graph below. So power dissipated would be around 0.77W; easily dealt with a small heatsink. (Also beware that TIP120 is not a single transistor package, but a Darlington.)