Lengua and lenguaje are the same when meaning "a verbal communication system".
La lengua/el lenguaje se utiliza para expresar lo que uno siente.
From there on they separate:
Lengua:
Tongue (the organ and synomym of language directly below)
Language (French, Spanish, etc. synonym of "idioma")
The language of a group of people or a particular period in time - La lengua de Cervantes, la lengua del medievo, la lengua catalana.
Lenguaje:
The way/style of expressing oneself: lenguaje callejero, académico, grosero, educado
Technical communication as in computer language - lenguaje informático
Other types of communication - the language of flowers (el lenguaje de las flores), the language of fans (el lenguaje de los abanicos), etc.
The use of or the capacity to speak - el lenguaje se aprende desde la cuna por imitación.
The suffix you mentioned -aje is often used to make verbs into nouns, as mentioned in a previous post - amerizar, amerizaje, i.e. to land on water, a water landing.
I'm sure my answer is not complete but hope it helps.
