Trying to get interest lower on a loan

Generally, lenders will not change loan terms "in the middle" of a loan contract. It causes legal ramifications when they modify the terms, and it's a lot easier for them to just say "no" than to deal with new disclosures, etc.

You may be able to refinance into a "new" loan with Citi, and then they might lower the APR on a new contract. However, by law, you can't refinance that loan with the same lender for at least a year.

How does that effect my credit report if I try to refinance with them?
Trying to improve my credit and don't want to many inquiries on it.

Unfortunately they will, most likely, pull your credit report to ensure that you're not undergoing any sort of financial distress.

I wouldn't worry about a single credit inquiry. It's when you are shopping for a car, a new home, and multiple credit cards at the same time - that's when lenders see potential for a problem with multiple inquiries.