I see no one braved a response
I read this a few times to see what you are after. You have a Blue ray DVD player?
Not a problem plugging the DVD direct to the TV with that connection, Unless you are planning to use the TV speaker as the center speaker having sound processed in the TV and the surround and other speakers by the receiver.
Or, the same issues if you plan on running the front speakers from the TV processing as I am not sure what you are driving with that receiver without the video feature.
Would there be any quality problems or video/sound sync problems doing it this way, rather than sending both the video and sound through an A/V receiver with HDMI?
No issues unless as stated above when you mix which component is processing what audio signals.
What are the benefits from doing it this way vs all through an A/V receiver?
Benefits are switching multiple video sources with one component, receiver. Receiver, some, will do video signal processing, up-conversion?
And also, how exactly would you hook it up to the cable box and DVD player? Would you use 1 Toslink cable from the player to the receiver, and then 6 Toslinks from the receiver to the 5 surround speakers + subwoofer?
Toslink cable is for digital audio signal only that has to be processed and routed to the right speakers. From the receiver, you use speaker cables, wire pairs of around 14ga-12ga.
You may be misunderstanding cable functions?
You would send the HDMI to the TV and a toslink to the receiver for the 5.1 processing.