If William Walker succeeds in establishing his independent...Republic of Sonora in its entirety), how might it affect things? Could it have still been possible for the US to make other deals with the Mexicans, such as the Gadsden Purchase?
Nearly all of the Gadsden Purchase was in Sonora, and would have been under Walker's control.
If so, Mexico would be more inclined to 'sell' it and also the larger area the US offered to buy, which extended to all Sonora and Baja California at maximum.
The question is whether the US would pay Mexico anything, or just try to deal with Walker alone.
And how would its existence- either as an independent bordering nation explicitly supporting the institution of slavery,
There's a problem. Walker's pro-slavery agenda would alienate northerners. It's possible that the difference might be split - the US supporting "Sonora" only if slavery was not introduced. And it's hard to see what use slaves would be in Sonora.
...or as a fully fledged member of the CSA- affect the scope, path and outcome of the US Civil War?
Independent Sonora would have trouble connecting to the CSA. I think the US would mount an expedition from California to get rid of Walker and his pro-CSA forces.
BTW, this could have a major knock-on for California politics in the 1850s. California, though a free state, was politically dominated by the "Chivalry", the pro-Southern wing of the Democrats. If there is a slave state of Sonora, many of the Chivalry and their supporters may go there, leaving California to swing Republican earlier.