Unilateral proptosis & bilateral 6th nerve palsy is seen in -
Cavernous sinus thrombosis

The cavernous sinus is a cavity at the base of the brain that contains a vein, several nerves, and other structures.
The vein and cavity run between the large bone at the base of the skull (sphenoid bone) and temporal bone (near the temple), therefore infection to cavernous sinus can spread from the sinuses, teeth, ears, eyes, noses or skin of the face. 

Routes of spread include
i. Ethmoid sinus (most common) via ophthalmic veins
ii. Sphenoid sinus by direct spread
iii. Frontal sinus via supraorbital and ophthalmic veins
iv. Orbit by ophthalmic veins
v. Upper lid via Angular vein and ophthalmic veins
vi. Ear by petrosal venous sinuses 
Clinical features of cavernous sinus thrombosis

Onset is abrupt with chills and rigor 
It is bilateral
Involvement of III rd, IV th, V th and VI th cranial nerve
Chemosis of conjunctiva
Proptosis of the eye with limited movements
Papilloedema
Pupils are dilated and fixed
Decreased vision
Decreased sensation in distribution of Vth nerve (ophthalmic division)
Progressive ophthalmoplegia (specially for lateral gaze)
Cavernous sinus thrombosis
Meningitis
Hydrocephalus
None