Lengua means tongue but it also means language in some cases.

Lenguaje could be translated as language but i think its most accurate translation is speech.

Idioma is the best Spanish word for language. (not to be confused with "idiom" which is translated as "modismo")

Examples:

Enseñame la lengua = Show me the tongue

Habla cuatro lenguas = He/She speaks four languages

Tienes que mejorar tu lenguaje = You have to improve your speech

Habla cuatro idiomas = He/She speaks four languages

lengua = tongue, language
lenguaje = language
idioma = language

Yes, precisely. "Mother tongue" is "lengua materna".

You have to shift the 'u' somewhat to the end: "tongue"

Deconstruct hollowgasts until the rooster laws in florida come home darlin.

If you spelled tongue like you did in the post, that's probably why. The u is too far away from where it should be for it to be recognized as a typo.