ok I found a little strange settings for inventory content. It's also part of "Organic Groups" and this overwright your permissions.
1) Go to /admin/structure/types/manage/lab_inventory/fields and delete Group audience field from field list. This should remove relationship between organic groups and inventory content type. Then, the permissions will be based primarily on /admin/people/permissions page. It is a five rows - Create, Edit own, Edit any, Delete own, Delete any.
2) To see Administration menu for new user, go to Administration page with default administrator and click on "Members" from left hover toolbar. Add new user to Administration space as member.
Same siutation is with Calendar and Wiki space. To enable create new events or create new documnet page in Wiki add users as members to respective spaces in "Members" tab.
I solved this bug in development and add instructions how to use in Help. New release of FreeLAB will have implemented this features.
Thank you for sharing your experience!