This is so commonly misunderstood that I shall attempt to provide a short précis, then link to a good explanation which is not overly-technical.
Common misconception - earthing/grounding
Most small household appliances are not, in fact, earthed at all - even if there is a valid earth pin on the mains plug.
They are what is known as 'double insulated' & carry this logo
The 'transformers' on this type of equipment are not the same as in high-power equipment & don't use a step-down coil transformer, linear supply. Instead they use a switch-mode supply.
All PSUs, whether linear or switch-mode leak a small amount of current due to capacitance. In double-insulated devices there is no true ground/earth so the design is optimised to try to prevent this leakage.
Unfortunately, switch-mode supplies run at very high frequencies, so the smaller capacitance in the smaller transformer ends up leaking just as much as the larger capacitance in a larger transformer running at 50 or at 60Hz for example.
The leakage current is usually less than 1mA and is often as low as 10uA, but even at such low current, it can still be present and still can cause problems.
So - the tingle or buzz that you feel is in fact you becoming the temporary ground/earth for this circuit.
It is usually safe, & any reputable manufacturer will ensure their PSUs are well within tolerance.
Cheap copies are cheap because they don't bother with most of the safety circuitry… you know, the stuff that makes the expensive one expensive in the first place ;-)
Have look at http://www.unitechelectronics.com/sparks.htm for a much more detailed explanation. Though it is aimed at hi-fi users & deals in 240v mains, the basic principles are identical.
The MagSafe board is a very easy repair for someone who has patience and a small amount of training fixing any computer at all. The MacBook Pro aren't as fidgety to repair as the iMac and MacBook Air (where I would say you should be a skilled technician with official training materials available to work on them reliably).
You're looking to replace this with a handful of screws turned.
Go to iFixit.com and look up the repair guide and parts list for the DC in board and you will know if you have the skills to find the right part, the right tools and the right manual to undertake this work. Otherwise, pay a local technician to do the job/source the parts/give you what they feel is a good guarantee. As for the adapter, buy a new or used one on the aftermarket or from Apple. When the cord is crushed and frayed, it is designed to fail in a way that won't cause a fire hazard. I wouldn't mess with that end of things or with repairing the transformer itself.
Best Answer
There is a product that also can help with this.
http://www.snuglet.com
It works by using a tiny machined piece of metal inside of the port on the computer to strengthen the connection of the magsafe 2 connector.