mbam log says that it has detected explorer.exe file as virus. but it isn't. go to mbam quarantine and restore explorer.exe. you will get back your explorer.exe.
same thing applies to system.exe . but extract it to some other folder and upload it virustotal.com and post link here. mbam guys have some problem with trojan.agent detection. so first give us the link and will let you know what to do next.
you said that you have to start explorer manually from task manager. how do you do that. which executable will you execute through task manager. in which is folder is it? answer all.
nmb