I cannot reproduce the first issue that you're reporting. I've restarted my browser countless times since I've started using the plugin and my settings (incl. the private key) are preserved across restarts. Note that unfortunately (due to how the original Chrome plugin is designed) changes to the settings from the addon settings page (in about:addons) will only be persisted if you click the "Save" button directly below the checkbox "I have backed up my private key".
I've investigated the second issue, and have implemented a fix that is included in the next version (2.3.10 -- because I messed up submitting 2.3.8 and 2.3.9) which is currently in the review queue.
