Can rhinitis cause dizziness and nausea?

Possibly. Rhinitis is a non-specific term describing irritation of the nasal passages. It can originate from many different sources including allergic, irritant and infectious. Those agents can cause body responses like dizziness or nausea to a variable degree so depending on the agent any or all symptoms could occur together.
Possibly but rarely. If the post-nasal drips are severe, a few people may get nausea . If there is a sinus infection, dizziness may occur. However there are many more causes for the dizziness and nausea. Given that you also have dental pain, ask your dentist to check you out for temporomandibular joint dysfunction also.

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Flu shot 3 days ago, gas, nausea, runny nose, headache, mild chest pain, weakness, sweating/chills, and dizzy. If not flu shot then what?

FLU. Flu vaccine itself can give rise to mild flu-like symptoms as you mentioned here (much easier/ milder than flu itself). Drink a lot of fluid, take enough rest. Can have Ibuprofen or Acetaminophen for aches and fever (no aspirin). Hope you will feel better soon. Read more...