For broken minis, what Neal suggested works well, especially if you are using the goopy polystyrene cement. The best thing about the goopy stuff, like Testors, is it effectively melts the parts together like a weld and forms a bond stronger than the original plastic. You can also sculpt thick polystyrene glue to some degree.
For some models, you'll want Blu-Tack or equivalent to hold the mini in position while it dries.
For moderatley bent minis, what Vaaish suggested works great. For severely bent minis, I just cut or snap the bent part off and then glue it back on. It's just less of a headache than dealing with a bent part that is barely hanging on. On the upside, this gives you a shining opportunity to change the pose of the mini and add some variety to the stand.
I have not yet found an elegant way to get minis out of an Epic base. Honestly, I just grab sprue cutters and scissor around the well that the minis sit in. After that it is usually and easy affair to get the remainder of the base cut from the mini. It's far easier to find a new base vs finding new minis to replace ones destroyed while trying to save both base & mini.