1. Technically, yes, but letting lithium batteries go unused for long periods of time is bad for them in general. Keeping them in the case is on the balance probably better for them than the alternative of letting them sit outside the case since it'll take longer for them to run completely dry, at which point they won't be able to accept a new charge.
2. Some of the proprietary features like the touch controls or EQ settings do require a mobile app to configure, but apart from that you shouldn't have any issues connecting it to your Linux machine as a generic audio device so long as you've kept the Bluetooth stack reasonably up to date.