How to identify a faulty peripheral before connecting to my Beaglebone microcontroller? Board damaged connecting to ADXL345
1) Before connecting an external peripheral, how do i ensure it has no shorts? What apparatus or instruments i use to confirm that?


Please don't think about looking for "shorts". There are many ways that any electronic device can be faulty. Most of those faults are not "shorts".

If you really wanted to check for "shorts" then you need to be clearer what you mean - shorts between where and where?

In reality, you cannot test for most faults, before trying to use a module like yours.


  2) What should be the sequence of connection to beaglebone? Do i connect ground and Vcc at the end? Should i connect them when the board is turned off? Should ground connect before Vcc?


The answer is your suggestion: "Should i connect them when the board is turned off" Yes, make all connections with all power off. Only after all the connections are made (and you have checked that they are correct) then turn on the power. When making connections with the power off, the order of making the individual connections doesn't matter.


  Another mistake is, i connected first the beaglebone +3V to 3V3 of ADXL345. I believe now from arrow sign 3V3 is the output voltage of ADXL345 and VIN is the input voltage or supply voltage which powers on the device.


You need to look at the schematic for that specific breakout module, or read its instructions (or reverse-engineer the schematic yourself). The arrow does suggest it's an output, but it might also be usable as a 3.3 V power input, in some cases, if the "VIN" connection is not used - depending on the design of that board.