The elevator doesn't seem to be faster than the stairs. In most cases, it actually takes more time to walk to the elevator then to just use the stairs. This could be imporved if the game would automatically pick the faster option for each scenario.
I've been playing for several hours on and off during the evening, and I've all trophies except for killing the Shield King. I've not encountered him yet. Around what level does he turn up? I've not got past level 22.