So... first off, as I said in a comment, the requirements for practicing law varies from state to state. Suits takes place in New York, so we'll look at the laws there...
The requirements to take the bar in New York are a bit complicated but, assume, in general, one must have successfully completed at least one year (28 credit hours) of law school at an approved university in the US plus working as a clerk in a law office to total four years or completed a law degree outside the US or at an unapproved university in the US (plus 5 years of practice).
ABA Approved Law School Study (JD graduates) - Applicant attended and was graduated with a first degree in law from a law school or law schools in the United States which at all times during the period of applicant's attendance was or were approved by the American Bar Association (ABA). (Section 520.3 of the Rules of the Court of Appeals)
List of ABA Approved Law Schools
Law Office Study/Clerkship - A combination of law school study at an ABA approved law school and law office study (520.4 of the Rules of the Court of Appeals).
Unapproved Law School Study - Graduation from an unapproved law school in the United States with a Juris Doctor degree and practice in a jurisdiction where the applicant has been admitted for 5 of the 7 years immediately preceding application to sit for the New York bar examination. (Section 520.5 of the Rules of the Court of Appeals)
Foreign Law School Study – Successful completion of a program of study at a law school outside of the United States that is both durationally and substantively equivalent to a program of study at an approved law school in the United States, and if required, successful completion of an additional program of study at an approved law school in the United States. (Section 520.6 of the Rules of the Court of Appeals) (See also, "Foreign Legal Education" section of this website)
So, based on option 2 above (further explained farther down the page) no you don't need a law degree to practice law in the state of New York. Yes, you do need to pass the bar exam but you can't take the bar exam without meeting the minimum requirements, which, in three out of four cases, means having a law degree. You can't walk in off the street and take the bar after studying for it on your own.
Mike, unfortunately, doesn't seem to have ever entered law school in the first place. While he's probably met the requirements for working in a law firm for four years, he does not meet the minimum requirement of having 28 credit hours from an approved law school.
Mike has an encyclopedic knowledge of law thanks to his eidetic memory, but never attended law school or passed the bar exam, at least not under his own name.
He would not qualify to take the bar, and, thus, could never pass it and therefore, could never legally practice law in New York state.
Your other questions have nothing to do with M&TV...
Generally, an American law degree is a "JD" or "Doctor of Jurisprudence". It is technically a doctoral-level degree - though you can get a Master's degree in law, most do not*.
There is no set bachelor's track for a JD. You can major in anything you like. Many opt for Government or Political Science majors but it's not like medical school. You don't need any specific knowledge to succeed in law school.
*From what I understand, most people who get a Masters of Law degree are foreigners wishing to practice law in the US.
An LL.M. degree from an ABA-approved law school qualifies a foreign legal graduate to take the bar exam in Alabama, California, New Hampshire, New York, Texas, as well as in the independent republic of Palau.