After reading readme.html from webpagetest_1_6.zip, I’m not sure if it’s possible to set up WebPagetest so that it’s only on a Linux/Apache server, without the need for Windows.
Any hints on that?
Nope, can’t be done - it depends on using IE to do the testing so at a minimum the test system needs to be running windows 
Umm, not so good.
Are you familiar with any open source or free “competition” which would run without Windows?
I assume you don’t have an early alpha version which would work with Firefox (which could be started on Linux) 
Sorry, haven’t started the work to get Firefox working. Show Slow has recently added automation support which may be an option.