How long do inquiries stay on a credit report and how can they be removed?

From my understanding they stay on your report for two years.
Disputing Information Directly With The Creditor

This method is somewhat like writing letters to the Credit Bureaus, however,
the difference is that the dispute letters are sent directly to the creditor. One thing
to remember is that the department stores, banks, etc. have the power to take
anything off a credit report that they have placed on a credit report.

A number of things must be included in the creditor letter. Please give the
store your account number. State exactly what the problem on your credit report
is, and why it should be removed. Ask the creditor to verify the amounts.
Demand that they correct or remove the incorrect item.

Unlike the dispute letter method, this type of letter can be typed out rather
than written.