No you can't just buy any mosfet. Even if you found the exact chip, this is not something I would recommend but seeing how your PC is non-functional right now, why not mess around with it?
Unless its under warranty, Most likely you'll have to get a new lappy. With that solution looming, like you I would be inclined to see what I can do with the toshiba. Unfortunately we have no idea if that one mosfet will correct all your problems. But I wouldn't assume that a similar series of mosfet would fix your PC, there are prolly thousands of different little mosfets made for this type of application. I think its likely they would all have similar but not exactly the same voltage/current requirements.
The reason I believe you'll have a difficult time is, well have you ever tried to find a replacement power supply for a laptop? If you did, you'd find that they are rated differently for each model, they are all over the place and there is no standard volt/amp requirements for these things, I hope this answers your question, or at least prepares you for the possible outcome.
BTW, I've found that replacement motherboards are more expensive than a brand new machine.