Is it supported in any way to use HERE Maps APIs on client side without using the HERE maps services? Possibly by providing our own server and maps for the API to render on the Android device? If yes, could you tell me where can I find some sort of documentation that could get me started?
You can use HERE together with your own map data. Both, the HERE SDK Navigate Edition and the Explore Edition allow a headless mode. It's described here.
Scroll to the chapter 'Use Engines with or without a Map View'. Basically, you can use the HERE SDK and all exposed HERE services like Search, Routing, Traffic, and so on, in a stand-alone way. If you wish, you can fully do the maps rendering on your own and e.g. use HERE on top to search for places.