With non-instruction, I mean things like exceptions, extended features(like VME, PSE, PAE, debugger registers, Paging TLBs, TSS-related faults at various stages and during interrupt(IDT) TSS handling, protected mode behaviour(all exceptions possible when running any configuration, even stack faults in various documented modes and stages of execution(like halfway during a fault handling's processing), values pushed on the stack for the various stages within exception handling, halfway processed faults(for example faults that happen after other checks already passed during interrupts etc., double faults etc.).

Is there any testsuite that checks all of those?

test386.asm checks basic instructions mostly, but barely any of the above?