Does obtaining a long-term stay visa to a country in the Schengen area enable me to travel freely within the Schengen area?

Yes, you can travel anywhere within Schengen area (but not the EU). Before travelling, be careful to check which countries actually belong to Schengen as Schengen is not the same as European Union. There are countries, like United Kingdom or Ireland, that are members of the EU but opted out from Schengen. To enter these countries, you will need new visa. But when it comes to your question, once you obtain valid visa for any country in Schengen area, you can travel freely within Schengen even without having your passport or ID card checked. The borders between these countries have been removed to support free trade.

The bureaucrats that created the Schengen zone arrived rather arbitrarily at the 90 day figure, which ends up being a problem for the semester-length period of study that is usually 15 days longer than 90.  So how do you best handle that?  Easiest way is to spend those extra two weeks in the UK.  It would be nice if the pencil pushers could address this issue that impacts thousands each year, but I've never know a bureaucrat yet who was all that concerned about getting it right.

A long-stay national visa (D type visa) issued by a Schengen State for the purpose of studies, would allow you to move freely for up to 90 days in any 180-day period within the territories of the other Schengen States without requiring a short-stay (C type) Schengen visa. It is clearly stated in Article 21 of the Convention Implementing the Schengen Agreement.