Traumflug

It's safe to delete image types not needed. However, there might be modules using some of these with direct links. A developer would use something like 'grep -rn thickbox_default modules themes' to find these direct links. Links created with PHP methods (should be done everywhere) are safe.
Another approach is to simply delete them and look whether the shop misses some images. One can also restore like-named image types by simply adding them.
Yes. Also watch out for image folders with nothing but an index.php file inside. One can delete these as well.
 
All kinds of images are stored there. One can delete them, they'll get recreated.
 
I think so. Fairly easy to find out by simply trying. Look at image file creation dates.
@zen wrote some "image laundry" code about a year ago, which does all such cleaning as a background task. But this didn't make it into the distribution, yet.

It does.
Even if it didn't, thirty bee's updater compares distribution files with those on disk and reports differences. It also detects files on disk, but not in the distribution.

File manager isn't a back office controller, so it can't get put into a menu item.
However, wherever one can edit page content, the editor for this field has a button to open this file manager.

See themes/niara/sass/partials/_footer.scss, change it, then rebuild the theme.
If you don't want to rebuild the theme, see #footer in themes/niara/css/global.css

This should happen already. At least with discounts set in back office -> Customers -> Groups -> (edit).
One can look up back office -> Customers -> Customers -> (click), in one of the panels is a list of groups a customer belongs to.

Standard migration method is to upgrade this shop to PrestaShop 1.6.1 using PrestaShop updates, then migrating it to thirty bees.
If this doesn't work out, there are modules and services which don't migrate the shop software, but copy products, customers, orders and all this stuff from an old to a new shop. For example these:
https://store.thirtybees.com/shop-modules/administration/migrationpro-migrate-from-prestashop-to-thirty-bees-migration-tool
http://www.shopping-cart-migration.com/#a_aid=thirtybees

Obviously some trouble with module cronjobs.
Very odd table name. And looking at this database upgrade script: https://github.com/thirtybees/cronjobs/blob/master/upgrade/upgrade-2.1.0.php#L32, somebody apparently forgot the table name prefix.
Regarding shop recovery, is the shop actually broken? I'd try to go to back office using a new browser tab/window, it should work.
To recover the module, it might be necessary to execute this one line in the upgrade script manually, using phpMyAdmin and the actual table name.