F.Wolf, from the diagnoses given by you, the Medion laptop could have just sustained a Power Circuit Board damage, or it may be worse, and could be the motherboard.
The only way to find out is to open the laptop, (ALL power removed. AC adapter and Battery), and;
1) Physically inspect for visual damage to the Power Circuit Board, and the motherboard.
Not a very good diagnoses technique other than for apparent damage.
2) Buy, and install a Power Circuit Board.
3) Next in line is to replace the motherboard.
Due to the cost of just the Power Circuit Board alone, I feel this is redundant.
Also borderline redundancy, due to I hate just changing out parts to see if it cures the problem.
Quick search on the internet for a Power Circuit Board, for the Medion MD96418,http://www.mattotechinternational.nl/E_frame.html?http://www.mattotechinternational.nl/Powerboard+Medion+MD96418++E_art_197.html
(As you can see the DC Power Jack is mounted on the Power Circuit Board, and the Power Circuit Board is a separate circuit board from the motherboard.
It could BE just a bad Power Circuit Board)
For the Power Circuit Board cost alone, I feel you're well on your way towards the purchase of a decent used laptop.
Which brings us to removing the Harddrive, installing it in a decent, affordable external enclosure, and obtaining the information off of the drive.
I haven't repaired a Medion MD94600 series of Notebook PC.
I can't find at present a Service Manual online to give to you for disassembling the laptop, nor can I find the specifications.
Therefore I cannot tell you if your need a 2.5 IDE (PATA) external enclosure, or a 2.5 SATA external enclosure.
( A laptop harddrive is 2 and 1/2 inches across in Width.
The drive will have IDE or SATA marked on it, or you can use the Model Number online to look up the specifications.
This is one example of a decent (IMHO) 2.5 IDE external enclosure, (PATA)http://www.tigerdirect.com/applications/SearchTools/item-details.asp?EdpNo=1086960&CatId=2782
This is one example of a decent 2.5 SATA external enclosure,http://www.tigerdirect.com/applications/SearchTools/item-details.asp?EdpNo=3195233&CatId=2783
[ You may find information here, to help in disassembly to replace the Power Circuit Board, should this be the direction you wish to take,http://repair4laptop.org/disassembly_medion.html