WordPress doesn’t use sessions at all so it isn’t going to help you “control your sessions”. You would have to incorporate the code yourself.
SSI? Server Side Includes? WordPress also doesn’t use Server Side Includes, and neither should you. They technology is ancient, bug prone — also known as susceptible to being hacked — and very, very limited. WordPress uses, like pretty much all modern web applications, a server side scripting language to generate the page markup. WordPress uses PHP in particular. PHP does make it easy to separate page components into different files, but this is not the same as SSI.
I am not sure exactly what you mean by “…plug it into a site…” but probably not. Whatever you have already built would have to be converted into a WordPress theme to get WordPress involved in managing file includes.
Yes, you can use WordPress’ backend without any of the front end components– thus having a kind of “bridge” to the database. But don’t. WordPress is a heavy application if you aren’t using most of it. It is a pretty quick, decent piece of code if you need it, but if you just want a connection to the DB it is much, much too heavy.
Answer: commit to WordPress or roll your own CMS-ish thing to do whatever it is you need.