Push on button. Camera turns on, lens pops out.

Most of these kinds of errors come as a result of some kind of impact, sand or dirt exposure, possible water damage, or some other problem with the lens.
There is a possibility that the lens is off track.
Try pulling or pushing (not too hard either way) and see if it may be off track and return it to the correct setting.
Since it is not working anyway, try the above a few times and then test the camera to work right.
I have fixed more than one with similar (off track problems) and they worked good after wards.
Worth a try.
If nothing works, I suggest to sell it (include all that you have with the sale) on a site such as Ebay as parts, or as is repair. It should bring something like $20.00 to $50.00. The LCD is worth something over $15.00 alone. The other parts of the camera has value to those who repair cameras.