Is it possible to place plugin files inside a WordPress theme?

No. However you can easily prevent them from updating by either increasing the version number of the plugin (not really recommenced as it’s nice to have a point of reference), or by editing the plugin name – Example Plugin will get flagged for an update, but Example Plugin (edited) won’t.

If you’re comfortable enough with editing plugins, and the plugin you’re editing conforms with WordPress standards you should use its plugin hooks. This way you can keep your custom code, and have an up to date plugin. Give this a read for more information on hooks