PS: when I use my friend's magsafe it
charges my MacBook Pro.
Sorry if this is too obvious, but if your friend's magsafe charger indeed charges the battery without any problems, then it certainly seems like the problem is in your charger. (Somewhere in it; the cord, magsafe adapter, or some other part.)
How do Powerline Adapters Work?
Powerline Adapters are, in effect, baluns1 that embed an RF frequency on electrical wire. What they are doing is taking the unbalanced CAT5 signal (which requires a ground connection) and converting it to a balanced RF modulated signal that is transmitted over the building's copper electrical wiring. At the other end, it does the opposite to convert it back to CAT5
Why should this affect your battery?
The powerline adapter doesn't affect your battery per se. What affects your battery is how much power is required to make your connection and how much traffic you send/receive. The more power required, the less life on your battery.
Ethernet (CAT5) is designed for "high speed, long distance, and high reliability (noise immunity). All of these features generally require trade offs in power consumption."2 Additionally, if the SNR (Signal to Noise Ratio3) is high, it will increase power and/or lower transmission speeds to accommodate
WiFi is basically the same but with the benefit that it will adjust its power based on the quality of the signal unlike Ethernet. Where your signal is excellent, your WiFi adapter will lower its transmission power as extra power would simply be a waste. Where the signal quality is poor, it will increase the transmission power to ensure connectivity.
This is also why your cell phone's battery also doesn't last when the signal is weak.
Conclusion...
In your scenario, you may have a WiFi signal that is excellent and for the work that you do, you don't see an impact in terms of speed or reliability going over WiFi. Your upside is longer battery life.
However, should you have the need for higher speeds (like "moving" vast amounts of data), WiFi simply won't keep up with the transmission requirements. CAT5 will be the advantage here. Whether it's on a power line adapter or on a switch will not change the power consumption of Ethernet.
1Balun - Balanced/Unbalanced. A device that links together dissimilar wire types and attempts to minimize any negative effects to the signal that would normally result from the dissimilarity. Dictionary of Networking Terms
2 Microchip - high power consumption in ethernet mode
3 In an electromagnetic signal, the ratio of the amplitude (strength) of a signal to the amplitude of the ambient radiation and other signal disturbances that are present, usually expressed in decibels (dB). Dictionary of Networking Terms.
Best Answer
If the apps are freezing then the Mac is trying to use the battery for power and failing.
Most (if not all) portable Macs use both AC and battery for short periods (like opening apps). A bad battery can cause crashes and data loss as the Mac tries to use the battery and cannot.
The CPU throttles without the battery because the Mac sees the missing battery and lowers power use. Your Mac has freezes because you exceed the AC adapter slightly. I assume a larger power demand will cause a crash.
At the very least the battery can swell and bend the Mac, and can sometimes catch fire if the defective cell cannot turn off. It would be safer to remove the battery and use on AC.
With only 3 cycles the battery looks like it was constantly full for 8 years (non-retina device) so it is time for a fresh battery. Batteries need to be used occasionally.