That field syncs it from the server to the client.
Makes sense since the delta movement is not necessary for syncing, only the position and rotation.
You can't, lerping is not supposed to be done on the server anyways as it is meant for handling smoother animations on the client when rendering to the screen. You should just be using the raw position values if you want to use them.
Once again, player movement is handled on the client. The only information synced to the server is those that need to be sent to all players, which in terms of movement can typically be boiled down to (as an oversimplification) position and rotation. If you want other values, you'll need to sync them yourself using a packet. Using Entity#hurtMarked will allow you to sync the values you set on the server back to the client afterward.