Base System Device has no drivers installed; this sounds like a serious hardware issue. Is this a custom built machine?
Explain why all the Adobe entries in the Host file are there please? ??
You did not answer my questions.
Have you looked into the Base System Device / driver issue as this may be what is causing the crashes on reboot? Does the system crash on reboot normally or only when certain routines want to run during start-up?
Your logs do not show any major malware but there are some things that could be cleaned up. However, cleaning will involve shutting down processes and rebooting the system so making sure that can happen correctly would be helpful.
You are not on trial here but we would appreciate your removing Adobe or at least resetting your Host file.