Answer: Cavernous sinus.
Step 1: The combination of proptosis with bilateral sixth nerve (abducent) palsy localises the lesion to the cavernous sinus.
Step 2: The abducent nerve runs through the cavernous sinus. A cavernous sinus thrombosis, which can spread across the midline through the intercavernous sinuses, therefore produces bilateral abducens palsy.
Step 3: Venous congestion from cavernous sinus thrombosis causes proptosis, starting unilaterally and often becoming bilateral.
Why not the others: Orbital cellulitis, orbital pseudotumor and orbital lymphoma are confined to one orbit, so they cannot explain a bilateral cranial nerve palsy.
Ref: Parson's Diseases of the Eye, 22e, p. 497.