I find that if a worksheet (probably any child window will do) is created at startup (Tools-> Options-> Open/Close-> Start New Project: Worksheet) then Project Explorer will open at startup if it was open when you closed the previous Origin session. It behaves like the script window and Results Log in that respect.
No way that I know about... after all, why would you need Project Explorer if you have no child windows?
For similar reasons I still control startup with an autoexec macro that creates a worksheet and then minimizes it to get it out of the way. Add this to Macros.cnf...