Exhaust fumes seeping into our 4th floor apartment at night through bathroom and shower vents/fans
PLEASE seek professional assistance immediately.

There are two main issues with exhaust that can make you feel ill like this: Carbon Dioxide and Carbon Monoxide. Both are harmful, if not fatal, if the percentage present in the air you breath is too high. You DESPERATELY need to install a carbon monoxide detector, and contact your local housing authority. This is not something to be taken lightly, nor fixed by amateurs. HVAC follows particular code and regulations in many countries. I suspect that your venting exhaust is either too close to the garage venting, or is venting INTO the garage.