Hello,
1 - There is no option for that in HikaMarket and unfortunately I don't think that it would be possible using the HikaShop mass-actions (because it would require a filter for the "front-end context").
2 - A product can be added into an unpublished category yes. It's a good way to group your products for specific process, like discounts, custom fields, etc.
Regards,
Jerome - Obsidev.com
HikaMarket & HikaSerial developer / HikaShop core dev team.
Also helping the HikaShop support team when having some time or couldn't sleep.
By the way, do not send me private message, use the "contact us" form instead.