I think most people use laptops nowadays. I don't know much about it specifically but you could research a program called Abelton Live which is designed for live applications in particular. Other programs may do similar things but their response to pressing play may not be very good. I think you can tap a tempo and count in with Abelton.
Not sure about a click track. most logical way I can think of is to output the bass to one channel of the stereo image and the click to another, and only send the click to monitors.
If you don't have/don't want to use a laptop theres loop machines available. Not sure about ones with sequencing capability, but you could record a sequenced track as audio and trigger it in real time. Take a look at Boss RC series or Digitech Jam man amongst others. I have the Boss RC20 though - audio quality isnt high and only has 16 minutes of sampling time or so.
Other than that I don't know. There may be something MIDI based more suited to your needs. Final idea would be to do a Nick Zinner from the Yeah Yeah Yeahs - buy some more amps like Fender Bassman or similar, and split your signal up to fill out the sound.
Andy