How to have safari keep me logged in on all sites?

... I've done a fair bit of research and understand that i need to use cmd+q to quit safari instead of the red x in order to be able to continue from my last session.

That seems to be the opposite of what you intend. If you don't want to be logged out, don't quit Safari. The behaviour of many OS X programs including Safari is such that clicking the red button merely closes the window, which is not the same thing as quitting the program.

Some sites, such as this one, will log out upon quitting the browser. Other sites having a "keep me logged in" feature may log in to your account automatically through the use of cookies. If a site is logging you out even when you don't quit Safari, the reason is that the specific site is designed to do that. Circumventing that annoyance would require a Safari Extension designed to periodically refresh the page, for example.

No. Restarting quits Safari so the same thing will occur. The only way to remain logged in is to use a website feature that logs you in automatically through the use of cookies — if the website offers it — which isn't really the same as remaining logged in anyway.