If I need to uninstall and re-install Timetracker, are my records and data safe? How are work items changed with the installation or uninstallation of Timetracker?
Yes, the recorded time data is safe.
Uninstalling Timetracker for TFS does only remove the executables and system registrations. The data is stored in the database and not touched by the uninstallation. If you install Timetracker, setup checks if the tables exist in the database. If there are no tables, the tables are created. If the tables exist, they are left in their state as they are. If the tables exist in a scheme of a previous version of Timetracker, the tables are updated.
It is similar with Timetracker for VSTS, the only difference being that you don't need to run any installer; you do everything within your VSTS account.
Fields that were modified by Timetracker (like Completed/Remaining work) will keep their values as this data is stored directly in VSTS and TFS and Timetracker only modifies them. After uninstall, these fields will behave as usual; Timetracker will not modify them.