My google-fu and forum search skills are failing me. Where are the rules for “social combat”, using strain in place of wounds and social checks instead of weapon rolls?

Also, do the RAW contain things like equivalents for “weapon” base damage (maybe using the relevant characteristic), Willpower as Soak, etc.? Any good houserules that fill in gaps in RAW?

There are no defined rules for social combat in any of the Star Wars supplements, unless they've printed something I'm not tracking.

However, they did print a semblance of such rules in Genesys, their brand free version of the narrative dice system. I believe pages 118-123 in that core book are what you're looking for.

Genesys rules are basically:

1. As an action, you can make an opposed Social check using the regular rules.
2. On a success, deal 1 Strain per S. On a failure, suffer 2 Strain.
3. Once the target is at >50% Strain, they're willing to compromise.
4. Once the target is at >100% Strain, they capitulate entirely.

The bit that makes it interesting is the way boost/setback dice are allocated by how the PCs skills and arguments play off the NPC Strengths/Flaws/Ambitions/Fears.

Adapting Genesys is probably best for pure mechanics, but there are useful very guides on social conflict in Far Horizons and Desperate Allies.