Are SOT exempt from WA laws on SBR/Post 86 MG?

Yes a FFL/SOT in WA state can have Post 86 machine guns.  They can make their own "posties" or transfer post 86 samples which is most cases requires a LEO demo letter.

What happens to the NFA weapons owned by a SOT if the SOT's license lapses?  Do they have to be surrendered, sold, or how are they to be disposed of?

Typically and practically, they're turned back into kits and sold to the next SOT.

SOT's can have SBR's, SBS on a ATF Form 3
SOT's can have Pre-may 86 MG on a ATF Form 3, no "demo letter" required.
SOT's can have Post 86 MG but they require a "demo letter" (also called a "love letter") from a Chief of Police on the specific MG's they want to look at.

FFL 07 with 02 SOT can build machine guns.
FFL 01 with 03 SOT can not build machine guns (the 07 means your a manufacturer, 01 is a dealer)