I'd like to know if there's any way to see logs of the screensaver activity. Something like that:
02/02/2013 10:16:47 Screensaver started 02/02/2013 10:17:23 Screensaver stopped
I'm running Win7 Home premium 64 with ElectricSheep screensaver.
Yes there is! However screensaver invoking and dismissal (event IDs 4802 and 4803 respectively) aren't events that are normally logged by default, and so you need to turn auditing/logging of those events on first using the Group Policy Editor:
See my answer here for more details.
Edit: Since you're using Win7 Home Premium which doesn't include the Group Policy Editor, I tried using Process Monitor to figure out exactly which registry keys were being modified when auditing for these events is enabled/disabled. Unfortunately there were just too many changes being made and even after filtering I was unable to narrow it down sufficiently. Perhaps someone can help with this information, but if it's not possible the next best thing might be to just follow the instructions in this How to Enable “Group Policy Editor” (gpedit.msc) in Windows 7 Home Premium, Home Basic and Starter Editions? article.
I would look into the System Event Viewer. You can get to it by going to Start, Control Panel, Administrative Tools, Event Viewer.