Do all microSD slots support the current maximum capacity in the market, or is it device-specific? I know that currently (Dec 2013), microSD capacity is up to 64GB.

However reading some devices' specs, it says "includes microSD card slot, up to 32GB", is it a device restriction, or it's because it was written before 64GB cards were available?
That will most likely depend on the device, a generic answer valid for all of them cannot be given. For example, the Samsung Galaxy S2 specs give 32 GB as max, still some user was able to get a 64 GB card working (see: 64gb SDXC Cards in Galaxy S2). For some older devices, the 32 GB limit might be a fact, while again others might just have specified it to be "on the safe site".

TL;DR: Yes, it's device dependant, some devices have their limits.