Sounds like your fan might be going out, the timing of it cutting on after 10-15 mins which is usually when you're laptop would be warmed up and it being a humming sound gives me that impression. The bearings for the fan might be done and it running bad or not running at all would cause the overheating and freezing you mention. If you can take the laptop apart it wouldn't cost too much to fix it, you can either add couple drops of oil to the fan bearings or just swap the fan out and it'll save you from buying a new laptop, so long as it hadn't ran soo hot to cause any permanent damage to any parts you should be able to fix it.
Some example pics of how to re-lube the bearings of a computer fan:
Being a laptop your fan will probably look something like this when you open it all up:
If you can't get the fan fixed check ebay for a replacement fan, shouldn't be much.
And here's a site that has step by step how to's for fixing laptop cooling/fan issues for a number of laptops, check to see and they just might have yours listed:
http://repair4laptop.org/notebook_fan.htmlAnd for the oil, any household 3 in 1 lubricating oil like this would work fine:
Hope that helps!!