Is it legal to not pay any income tax from my paycheck? As in, my employer withholds $0 from my paycheck, and I then pay all taxes due when I file my taxes? Thank you for any insight!

Claiming to be exempt from tax on your W-4 when you are not exempt can result in fines and underpayment penalties.  Change your plans.

 By the same logic I should receive additional money from the government as a penalty for over taxation. 

You asked a question pertaining to the way the tax law works.   You got an answer--not an opinion inviting debate about the "logic."

I didn’t dispute your statement. Just pointing out the inequity of the system for anyone else who reads my question and your factual answer.