There's a claim that rare earth minerals cannot be reclaimed from recycled electronics. Is that true?
Depends which elements you want to recover. The main problem is the economic aspect of recovery - this is due to the fact that one has to collect the materials from a multitude of electronic devices and bring them through a chemical purification process. Rising rare element prices might change this in the future. Addendum: quick google search gave http://www.molycorp.com/ as a developer of recycling technology.
They can't. The mining of rare elements isn't constrained by resources, only cost. Setting up the kind of infrastructure needed for REE recovery would always cost more than mining, so if the prices start going up again, mining will just be expanded.