Is there a problem with closing the lid of a MacBook Pro when using an external monitor?
Clamshell Mode

What you are describing when you operate your Mac laptop (Macbook/Pro/Air) is called closed clamshell mode or closed display mode.

In this Apple Support Article:  Use your Mac notebook computer in closed-display mode with an external display it goes into detail on what the requirements are and how your Mac will function depending on the version of OS X you are running.

But basically, you can hook up an external mouse, keyboard and display and use it like a desktop.  As far as what you can do with your Mac, pretty much anything you can do with it open, you can do with it closed

Ventilation

The vents on your Macbook are located in the rear and along the bottom - they are not obstructed when the lid is closed.

How heat gets dissipated is more determinant on where it's placed rather than if it's open or closed...


  Where you use the system may affect fan behavior and heat as well.
  Using the computer on a soft surface like a couch, pillow, bed, or
  your lap can trap heat requiring higher fan activity. Using the
  computer on a hard flat surface like a table or desk allows for
  optimal radiation of heat.


The vents are located along the back, not under the keyboard: 


  Mac computers have air vents that allow heated air to exit. The vents
  are in the back of the computer on the MacBook, MacBook Air, MacBook
  Pro, and Mac Pro computers. iMac (Mid 2012 and later) models vent
  along the bottom edge of the computer. Make sure the vents remain
  unblocked to allow your computer to perform at its best.


There is also no need to continually monitor the temperature of your machine.  The only time you need to do this is if you are having a problem like it's slowing down significantly or shutting down unexpectedly. 


  The System Management Controller (SMC) firmware on every Mac
  manages the thermal conditions of the computer regardless of the
  operating system.


Basically, if your Mac gets "too hot" it will automatically shut it down to protect itself.  But before that happens, it will slow everything down to a crawl.  

Source:  Apple Support - Learn about the fans in your Mac