1. lenguaje, lengua, idioma;
As far as I can tell, there´s no difference when it comes to a language. Obviously "lengua" can mean the "tongue" physical part of your mouth.
2. estudiar, aprender;
Well... you could study for a long time, and never learn! Or learn something without studying (to swim, for example). So, that's the same difference. Some things (languages, medicine, law) can never really be "learnt".
3. alquilar, rentar, arrendar;
They´re the same, although in Spain "alquilar" is more frequent, but i know that in other countries they use the other variants.
4. ascensor, elevador.
In spain we only use ascensor. Maybe in latin america elevador is more common, I don´t know for sure.