SQL Server Express averages 2-3% CPU but gets stuck at 100% sometimes
It's probably related to  your AutoClose setting.  Almost every resource you will find recommends turning this setting OFF.
Essentially it closes connections automatically, but it can cause a lot of issues since it's not done well.  It can and does cause all sorts of mysterious performance issues.
Here's a post from Buck Woody about the reasoning.
Here's a blog post from another SQL Server MVP about a situation where it caused issues.
Step 1 should be turning that setting off, then testing for a while.
There was a request to deprecate the feature but it looks like they will be leaving it in.