Hard to say if sound quality will be better or worse using your amps speaker level inputs, depends if the signals are transformer isolated or not(David Navones LOC's are).
One advantage of using speaker level inputs is that usually the amp will have a sensing circuit that will power the amp up when it see's an input signal thus avoiding having to run an amp trigger.
As far as Scoshe adapters go, if you want to break out the speaker level output then you need the TA02 and TA02R.
I ordered mine from a car stereo shop in Florida(
www.stereoinstallparts.com)
The details are in the Mod Shop in the Garage section of Tundra Solutions. The link is "How to Keep the Toyota Factory Stereo and Add a New Aftermarket Amplifier- The Basics of the Line-Level Converter."