I haven't made the first reservation yet in 2 trips there. We spent the night right outside the NE entrance at a NF campground, then stayed at Norris. The last time we got lucky I admit, took the last site available for the day at 11 AM. No electric, but there are flush toilets. I think you can do the same at the other CG's inside the park, though you have to be willing to try again the next day if need be.
If you need electric or more, then you need a reservation.
I think the 7 days is a good amount of time. It's a LOT of driving if you want to see as much as you can, wait for geysers, enjoy a picnic, etc. to be able to do it in 3 days. All depends what you want for your own enjoyment.