If a spatial database is configured to display maps through WMS, is it always configured to display through TMS and WMTS also? Or is this something that must be configured on the server side separately?
A spatial database by itself wouldn't render a map in response to a WMS request. That's where a web mapping engine comes into the equation. The web server recognises a WMS request, defers it to the web mapping engine. The web mapping engine pulls the data from the database and returns an map image. In a WMS request you can specify any size, bounding box and scale. That makes it really flexible, but every request has to go through all the steps I just described. That's fine for one client application or the end user is willing to sacrifice speed over real time data.
A TMS or WMTS is a tile cache: all map images are rendered at specific scales and specific sizes (typically 256*256 pixels). In case a client application requests a map tile through a TMS or WMTS request, the web mapping engine, or even the web server can respond with a pre-rendered iamge file without having to pull any data from the database! A tile cache is rendered through a separate piece of software, a tiling engine that actually makes requests for map images to a web mapping engine, for example using the WMS protocol.