My band has the same situation. Two guitar players and one keyboard player who are all writers and singers. They all write songs, and just bring them to the band, and we play them. Each guy takes the lead vocals on his own song, and the other guys sing back up / harmony. Some guys have a few songs, some guys write a lot of songs, and we pick and play the good ones that we all like. All three guys also pick covers that they do, and those have a huge range in style and quantity. All three guys have different styles, but they all sound like our band, because, well, it's our band playing them.
Like mellowinman said,,, bands now are so "genre-oriented." Don't worry about having a disjointed show... if the band plays good, and is developed as a unit, then it will sound like your band. If you saw the Rolling Stones and they played "Girl with Far Away Eyes" and later played "Miss You" would you think the band was disjointed? No. Totally different songs, different genres, but you would think it was the Rolling Stones doing their thing. You've got to sound like your band first. Then the material will all work together because it all has a common thread running through it... your band's sound. Just be sure to work the set-list so that the show has a good flow.
As far as "show equality", make sure everyone understands that the show comes first, and there is a time and place for all the material. In my band, if we have a 45min set at midnight at a rock club, we play the rockers, and that might mean that one of the guys doesn't get his slow ballad in that show, and gets less songs that one of the other guys. If we are playing a three hour private party, one of the other guys might not get to do his rocker because the other guy has a lot of more mellow music that is right for that gig. A band has to function as a unit where the whole is more important than the pieces. You give and you take in a band, and if you can't to that, then the band will decend into petty fights, and fail. (Unless you have a situation with one Band Leader who decides everything

)
In any given show my band might play dance hall country, rock-n-roll, a little punk, disco and dance, slow blues, southern rock, psychedelic-freak-out music, etc, etc... what ever we feel like playing that particular show. It's hard at first because people want you to be a "country band" or a "rock band" or a "dance party band." Well, we are just ourselves, we play what we feel, and it all comes out sounding like us... we love it, have a good time doing it, and people respond to that.
Good luck, OP!