I have done the 2 first steps:
1. I cleaned the temporary files
2. I run avast at boot level for the C and D disks. The external backup was not on.
The avast scan at boot level took a looooooooooooooonnnnnnnnnnnnnnggggggggggggggg time, and got 2 worms:
New Folder.0xe est infecté par Win32:Hakaglan [Wrm], Mis en quarantaine
LaCieSync_v7_1_028.exe est infecté par Win32:Trojan-gen {Other}, Mis en quarantaine
The Win32:Hakaglan [Wrm] was in several files.
I put all the files in quarantaine, and I could delete them, but my only concern is for the LaCieSync_v7_1_028.exe file, as it is what permits me to use the external backup disk.
I did not do the following steps yet, I am waiting for your advices about what to do with what I found till now.
note: I forgot to tell you that I have also on my PC PCtools spyware, and this one doesn't find anything, but I can do the other ones you recommand.