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###First question:

###Second question:

var MusicPlayer = React.createClass({           
    componentDidMount: function() {
        this.player = document.getElementById('audio_player');
    }
    ...
})

###Just some general observations/suggestions:

var MusicPlayer = React.createClass({           
    ...
    render: function () {       
        return(
            <div id="music_player">
                <Sounds />
                <Controls />
                <AudioPlayer />
            </div>  
        );  
    }
})

First question:

Second question:

var MusicPlayer = React.createClass({            
    componentDidMount: function() {
        this.player = document.getElementById('audio_player');
    }
    ...
})

Just some general observations/suggestions:

var MusicPlayer = React.createClass({            
    ...
    render: function () {       
        return(
            <div id="music_player">
                <Sounds />
                <Controls />
                <AudioPlayer />
            </div>  
        );  
    }
})

var Sounds = function(props) {
    return (
        <div className="scrollable_container scrollable">
            <ul id="list_of_sounds">
            {props.sounds.map(function(sound) {
                return (
                    <li className="sound_list_item">
                      <span className="sound_title">{sound.soundTitle}</span>
                      <span className="sound_length">{sound.soundLength}</span>
                    </li>
                );
            }}
            </ul>
        </div> 
    );
}

var Sounds = function(props) {
    return (
        <div className="scrollable_container scrollable">
            <ul id="list_of_sounds">
            {props.sounds.map(function(sound) {
                return (
                    <li className="sound_list_item">
                      <span className="sound_title">{sound.soundTitle}</span>
                      <span className="sound_length">{sound.soundLength}</span>
                    </li>
                );
            })}
            </ul>
        </div> 
    );
}

[1] https://facebook.github.io/react/docs/state-and-lifecycle.html

Additionally, I would look to simplify the way you set the next and previous sounds as the current sound, by simply setting the currentSoundIndex and using that to determine the mp3src prop that is handed down to the stateless components that require it in the MusicPlayer's render function. Using that you can reduce the complexity of the class functions and remove the need for a getSoundInfo function.

[1] https://facebook.github.io/react/docs/state-and-lifecycle.html