Synonyms

Nauseate means to cause someone to feel disgust or nausea. Sicken has the same meaning of causing illness or disgust. Soar means to fly high, Ruin means to destroy, and Ooze means to leak slowly.

Exude means to display or discharge something noticeably. Ooze means to discharge or emit slowly, which matches this meaning. Clear means transparent, Soar means to fly high, and Gay means cheerful.

Paramour refers to a lover, typically in an illicit or secret relationship. Lover directly corresponds to this meaning. Claim means to demand, Deep means profound, and Mock means to ridicule.

Promulgate means to make something known widely or officially. Proclaim means to announce or declare publicly, which matches this meaning. Abuse means mistreatment, Gay means cheerful, and Tall means high.

Burlesque refers to a form of parody or mockery, often in performance. Mock means to ridicule or make fun of, which captures the essence of burlesque. Annoy means to irritate, Avoid means to shun, and Vivid means intense.

Confidential means intended to be kept private or not disclosed to unauthorized people. Secret means kept from knowledge or view, which is a perfect match. Deep means profound, Hide means to conceal, and Gay means cheerful.

Subside means to become less intense or severe. Ebb refers to a decline or reduction in intensity or flow. Old means aged, Annoy means to irritate, and Push means to shove.

Nasty means very dirty, filthy, offensive in taste or smell; nauseating. Thus, option 3.  

Relic means an object, custom, etc. that has survived, wholly or partially, from the past; or something that has historic interest because of its age and associations with the past. Thus, option 4.  

Sanctioin means the act of a recognized authority confirming or ratifying an action; support; encouragement; approval. Thus, option 1.

Futile means something that could not succeed; useless; vain; lacking vigor or purpose; inept or ineffective. Its antonym is 'fertile'. Synonym is 'useless'.  

Surge means large mass of or as of moving water; wave; swell. Soar means to fly, sail, or glide along high in the air. Both seem to be synonyms.  

Catapult is an ancient military contrivance for throwing or shooting stones, spears, etc. It is also a type of launcher that provides the force to hurl an airplane, missile, etc. from a deck or ramp to provide an initial high speed. Considering this, option 2 seems to be the right choice.

Halo means a ring of light that seems to encircle the sun, moon, or other luminous body. Option 1 seems to be the perfect synonym.

Clamorous means loud and confused; noisy.