There's probably not one single documentation location that explains everything you want to know, but here are some links:
Information about hosted feature and tile layers
About hosted features and tiles—Help | ArcGIS
Feature layers support vector feature querying, visualization, and editing. Feature layers are most appropriate for visualizing data on top of your basemaps. In web apps, feature layers are drawn by the browser and support interactive highlighting, queries, and pop-ups.
Tile layers support fast map visualization using a collection of predrawn map images, or tiles. These tiles are created and stored on the server after you upload your data. Tiled layers are appropriate for basemaps that give your maps geographic context.
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About ArcGIS Server web services
ArcGIS Server web services—Help | ArcGIS
ArcGIS Help (10.2, 10.2.1, and 10.2.2)
The thing to remember is that different layer types support different capabilities. If you want super fast drawing, tile layers are the way to go. But you can't turn on/off individual layers.
The fact that arcGIS online can access an individual layer of an ArcGIS Server map service is a feature specific to ArcGIS Online. When you work with a layer like this, you are not accessing a dynamic map service, rather, you are completely bypassing the drawing capabilities of the server. What's happening is that the client (web browser, mobile app) is requesting the geometry (xy coordinates) of the features and drawing them as best it can. Most clients can't handle the complex symbology supported by the server. So, for example, if you have a road layer with complex line symbols that represent different types of roads, likely they would likely be drawn with simple colored lines. The post I added above which points to other forum posts attempt to explain the details. Also, accessing the layer via it's index is slower than accessing the tiles.
In general, I think ArcGIS Online requires that people rethink how they share out their data through web services on ArcGIS Server. In the past, you'd put all your layers into one service and make an application that used that service. Often, there was a one to one relationship between a web service and an application. Since there were only dynamic map services, there was never an issue about turning layers on and off. However, these weren't always the fastest to display.
Over time, people wanted faster display, like they saw in google maps. Well, google caches their maps as tiles and we enhanced the map service to support tiles. The nature of tiles is that they draw fast because the server has predrawn all tiles and when a request comes in, the set of tiles is returned. It's very fast, but since it's predrawn, you can't turn layers on/off.
So now, we advise people to think about how their layers will be used and organize them into services appropriately. In this way, services are more like data that are used in multiple maps. For layers that don't change frequently and are meant more as background information, like a basemap, you'd put them together and cache them. For layers that change frequently and are used independently across many maps, you'd make dynamic map services. We refer to these types of layers as operational layers. Whether you put more than one operational layer in a service depends on how you want to use it. If you want to use in separately, you'd put it in its own service.
Hope this helps,
Mike
