Can I just buy an intel laptop CPU or are they not sold for private users?
No more. Something like the 3730qm e.g. Doesn't exist anymore.
There are still very few manufacturers who make custom boards with bases for normal desktop cpu's. But it is usually very, very expensive.
CPUs in notebooks are, with a few exceptions (usually old and desktop CPUs), always soldered. So you can't just swap them, you would need a complete BGA rework station. That's why notebook CPUs are rarely available to buy individually
Laptop CPUs are now almost exclusively soldered, which is why I doubt that this is possible. You can do some research to find out which one is built into you. Either way: if you don't install an identical one, there may not only be incompatibilities with the motherboard, but also heat problems due to possibly different performance.
At least older CPUs (e.g. 2630QM) can still be found on platforms such as Ebay.
