Logic board intermittent problems a mystery

I had a friend find a macbook pro 13inch early 2011 in the trash. It starts up sometimes.

Here are the clues:

• When I connect the hdd and turn it on, the screen is black but the apple chime sounds, slowly, eventually the screen turns to blank white, then boot screen appears after 1 minute
• It has frozen at the point of the loading wheel, but more often loads all the way.
• A full startup takes about 4 minutes.
• After loading all the way it works fine but freezes occasionally requiring a restart.
• The keyboard backlight doesn't turn on until the OS has loaded
• When I disconnected the hard drive and turn the computer on, the fan runs really high, full blast until it is unplugged or turned off

I have a bad feeling that this is a logic board problem. If someone has a feeling as to what part of the logic board these problems might be caused by, that would be helpful.

Best case scenario, I hope this to be a problem with one part of the logic board so that I could replace that part.

 

That makes sense, any Idea what a good asking price is? As much as I can get?