The settings on the Backup Options panel are mainly for servers. If your computer is a personal one, you may leave them as they are.
A snapshot is an image of a disk taken at a certain time moment.
Well, backing up a large disk may be quite a lengthy process. As it is being done without stopping the system and all other programs that may be running during the process, it may appear that something write data on that part of the disk which has been already copied to the image. R-Drive Image correctly processes such situation by monitoring the process of writing data to the disk and re-writes the image file when necessary, but it is a good idea to minimize data writing during the process of image creation.
Snapshot providers notify applications that a snapshot is being taken so they may refrain from writing something on the disk, or, at least, minimize such writing. Sometimes that may slow disk backup, but that rarely occurs. That is why the options can be unchecked. If you don't have programs like databases running, you may leave these settings as they are.
R-Drive Image can stop creating the image and immediately notify the user when it meets bad sectors, or may continue working and notify the user at the end. This settings are important for corporate servers where the cost of data loss may be unimaginable. If you don't have bad sectors, you won't see those warnings.
AUX indeed means "auxiliary". These are applications that you may want to run before and after image creation. For example, you may want to stop some program, then clean your computer from some temporary files, then create an image, and then start that stopped program.