What is the difference between neuen and neues?
Haben takes the accusative. In the accusative, the -en suffix for adjectives is for male or for plural. Since its Das Fahrrad (Neuter), you have to use the -es suffix.
See here on declension of adjectives in German:
Why not "eines neues Fahrrad"?
Haben takes the Accusative. The accusative declension of "ein" for neuter (since its das Fahrrad) is "ein" (stays the same). Eines only exists in the Genitive, and it doesn't make sense as part of this sentence.
What about das Velo? I think it might be only used in Switzerland, but could it be added as a valid answer?