MacBook – Would the battery from a 13“ MacBook Pro (2010) work in a 15” MacBook Pro 2011

The built-in battery of your MacBook, MacBook Pro or MacBook Air is designed to deliver up to 1000 full charge and discharge cycles before it reaches 80 percent of its original capacity. Link.

Little differences in capacity are normal and are no indicator for a defect battery. Only contact Apple when your battery performes worse than the design capacity.

In the absence of a repair shop to run diagnostics, you might try isolating things in the following order:

1. Disconnect all external devices
2. Different power adapter
3. Remove RAM
4. Disconnect / remove the battery and then reset the SMC
5. Disconnect hard drive and optical drive

As you remove items, you remove causes of failure. The mac should still POST and beep if everything listed is taken away from the mac. Often, it's an odd part that is causing a failure of the system to charge or drawing too much power when these isolate the issue. In reality, it's likely the DC in board or main logic board, which means it's good you have warranty to cover that eventuality. Do use prudence in doing the above steps, you don't want to physically damage things and run the risk of warranty service being denied due to external damage or someone not technically capable doing work on your mac.

Even if you don't do all of the steps, you can rule out a few more things than you listed initially in the question. Also, Apple service providers will possibly be able to revert your firmware or at least confirm you have a working firmware install as well as be faster to isolate the cause than most consumers doing the steps listed above as they are set up to do this day in and day out.