With your battery issue, it seems as if your battery fails to hold a charge and probably needs to be replaced. How old is the battery? Even rechargeable batteries have a life span.
With your earbud problem it may be that the earphone socket in your phone is faulty. The connecting spring inside the actual socket may have lost tension or the socket may have come loose from the board.
To test which connection is actually faulty, plug your earbuds into a working phone socket. Take a note of which is the left earbud. Mark it somehow so that you you know which one it is. Plug the earbuds into your phone and make sure the left earbud is actually in the left ear. Now which side gives no sound?
Here is a picture which shows the connections for a plug
If it is the left earbud that is not working, have you fully inserted the plug? The plug should go right in until the holder touches the phone case
If it is and it is still not working or if it is the right earbud, here is a link which shows (not very well, also it is in Spanish but the video is self explanatory) how to dis-assemble a Blu Studio 5.0 (not your model I know but perhaps close enough to be of some help) so that you can gain access to the headphone socket. At 2:36 minutes into the video the headphone socket can be seen on the top edge of the phone (as viewed) from the right side (under the 'S'). You can then determine if it is loose and if not if it needs to be replaced.
If you do not have the tools or the skills to repair the socket, take it to a professional phone repair service to be fixed.