I was wondering if it is possible to run a browser (specifically a browser engine) on the server side. I do not just mean to render a page but to keep a browser open for some ti
Yes, it it very possible to run a web browser on a server. Ubuntu Server, for example can run firefox by simply installing firefox and xserver.
$ sudo apt-get install firefox
$ sudo apt-get install xserver-xorg
Then to get firefox running type:
$ X
This will leave you with a blank screen because there are no applications running on the display yet. You need to shift back to a new terminal with Ctrl-Alt-F2. Ctrl-Alt-F1 holds the X process now.
Login to the new terminal and set the environment’s display variable to :0 and launch firefox.
$ export DISPLAY=:0
$ firefox &
Firefox should now be running, but you'll notice it's got some quirks to it. If you decide that you want to go forward with this you should install a window manager. I'd go with a lightweight tiling window manager and run firefox and whatever other applications need graphical output within that window manager.