If a power cord shorts out, can it affect my Macbook Pro?

Question

I have my 2009 macbook pro 13" plugged in, then suddenly it turns itself off and needs to be restarted using the power button, and when it shuts down, the power cord seems to 'die' too. I think the power cord might be shorting out. Could this be? Could it possibly be a completely unrelated occurrence, where the cord dies and the computer separately spontaneously shuts off (perhaps the battery is dying)? No warning pops up or anything, there is just a tiny 'pop' sound and the screen goes black. I must then use the power button to turn it back on. When I turn the computer back on, the clock is set back to some time in 2000 and wifi is knocked out. But if I put the computer in sleep mode, close the lid, then open it, it reconnects to airport with no problem, and sometimes the light on the power cord will turn back on. I haven't been able to find a similar issue anywhere.
Some more background: the computer, when turned on and for example using Google Chrome and TextEdit, and these 2 applications only, the battery power quickly deteriorates. In addition, whereas the battery used to deteriorate down to the single digit percentages, it now tends to die at about anywhere from 10-30% battery life reading. This is where I am getting the idea my battery may be going bad. This is my 3rd or 4th power cord in 5 years.

Answer 1

This cord has not been wrapped or stored in damaging ways, I always circle it up, never use the little ears to wind the thinner cord on the A/C adaptor block because this has been shown to have problems, plus the cord gets all kinked up. I am as delicate as possible with my cords now because my previous ones have been damaged from wrapping/other technical issues (the first, which came with my Mac in '09, was the old 't' style and frayed from factory defect, the second stopped working after about 2 years so I was told it was probably due to age, the third stopped working spontaneously back in July of this year due to unknown causes, and this newly purchased one has been treated as described above, very delicately, wrapped in loose circles and placed in a padded carrying case). The cord was plugged into a surge protector when the issue occurred, at the home of a friend (not in my own home), when the issue began to occur.
Apple charges $80 for a subpar product that has time and time again failed for various and differing reasons, but they don't change anything because it makes them money when you have to keep buying new power supply cords every 6 months to a year or shorter. I have never had this particular problem where the cord overheats and shuts down my computer. I purchase from Best Buy because I can't afford to wait for 6 weeks while Apple ships new product, nor can I afford to pay extra for shipping. The nearest Apple store is an hour from my house, otherwise I'd go straight there and have my computer and cord diagnosed.
I am pretty sure the computer battery itself is dying, as well, as the unit is 5 years old.
I found it strange that no one else I could find had ever documented this same problem occurring.
Thanks for your time and advice.