Saw your logs. It is a difficult task to identify all the processes running on your system except some common tasks, unless someone knows what programs are installed on your system. Some advanced viruses inject the code in system files like the pagefile.sys or hiberfile.sys, which makes it very difficult to remove or to identify as most, if not all, antivirus programs do ignore these files when scanning for viruses.
I understand that you are unable to install some antiviruses. and probably this won't work either, but, I'll suggest give this a try.
1. Disable Windows Paging File (Also known as Virtual Memory). Control Panel > System > Under System Properties, select the tab Advanced >Under Performance, click Settings> Select Advanced tab and then under Virtual Memory, click on the button "Change", and then select the option "No Paging File". Doing this will remove the system protected pagefile.sys (one possible hiding location of the virus/trojan/worm).
2.If you have Hibernation enabled, Go to Power Options, under control panel, Click on the tab named "Hibernate" and uncheck the "Enable Hibernation" check box. This will do away with the other possible hideout of the virus (hiberfile.sys).
3. Once done with these, download a small utility named Unlocker from UNLOCKER 1.9.1 BY CEDRICK 'NITCH' COLLOMB and install it.
4. Now, try to unhide the hidden files from Folder Options. This will show you some hidden files and folders(Just be informed that there are a few important system files which are very important for system's integration, which, if deleted might corrupt your system altogether - Files named like ntdetect, ntldr, autoexec.bat, boot.sys, boot.inf, etc are a few) file as well as the autorun.inf file in your C drive. Try deleting autorun.inf, with unlocker running in the background and the process creating/accessing this particular file, would stop windows explorer from deleting this file and in a few seconds unlocker would present an option with the culprit process.
5. You unfortunately did remove Avast, whose Boot Time Scan comes in pretty handy in such scenarios. Even if it isn't able to run on the system due to the virus infection, you can actually schedule a boot time scan using the command line utility, by running cmd.exe as a user with Administrator rights, navigating to the location where avast is installed, usually C:/Program Files/Avast Software/Avast using CD command, then type sched.exe /A:* which specifies the application to schedule a boot time scan and /A: - defines the area to scan for the viruses, while * specifies it to scan all the drives.
6. Try installing Avast again, if possible, and schedule a boot time scan if you're successful in installing it. That'll help you avoid the pain of formatting your HDD.
, otherwise i'll try to get it downloaded later. 
