Regarding the key of a song, that can be a challenge when only viewing part of the melody or a bit of a song as in your example "Fair and Tender Ladies". If you can listen to a recording of the song in its entirety, it helps to get a better feel for the tonality as a whole, especially if you can write the chords and melody out. From there you can begin to identify the indicators of the key. Chris Lercher and Silver Light make good points of recommending the Last and First chords of the song, as does Jim Womack with looking for the most prominent chord. Unfortunately there's not one single variable to use to determine the key. A number of factors and indicators have to be taken into account and even in the end, two well trained and experienced musicians may differ on the final answer.
There's a nice video of the Carter Family performing the slightly different version on CMT http://www.cmt.com/videos/?artist=504715. It's in 4/4 instead of 3/4 and the melody is a little different, although it's close enough to know it’s the same song. The melody appears to be up a 4th from the manuscript you provided and the chords they're playing to accompany the melody are C, G, and Dm. C is both the first chord of each section (version, chorus) and the most prominent chord throughout. It also 'feels' to me to be in C, which is completely subjective, but one of the factors I use.
To compare apples-to-apples, we next need to transpose the chords down a 4th from what the Carter Family did, since their melody was up a 4th, so we can apply them to the melody you provided. The chords down a 4th would be G, D, Am. That would make G the first chord of each section and most prominent. Since the key signature has no sharps or flats, however, the key would be G Mixolydian (G-G in the key of C). Mixolydian, which is also Major/Ionian with a flatted 7th, is very common in folk music from the British Isles and subsequently the Appalachian Mountains and Southern US.
To your question about the instrument being in D - Many standard instruments are in different keys like the Alto Sax (Eb), Tenor Sax (Bb), Trumpet (Bb), Clarinet (Bb), and various recorders. There can be as many keys of each instrument as there are keys in western music: 12. Even though an instrument is in a different key, it can still play in all 12 keys, unless it’s a diatonic only instrument like some dulcimers. Instruments in different keys are much like cities in different time zones. People in each city can be a part of the same event around the world at the same moment, but for each of them it's a different time on the clock. Concert C could be considered GMT (Greenwich Mean Time).
So how do you figure out what key something is in that's written for one of these instruments in a different key? First determine the key of the instrument and what that key's relationship is to C. If the instrument is in D, then when a person plays what she/he considers a C to be on the instrument, it will sound the same as D on a guitar. If the instrument is in Eb, then when a person plays what she/he considers a C to be on the instrument, it will sound the same as Eb on the guitar. In order for different keyed instruments to play together, the music manuscript is transposed appropriately. For the instrument in D, the part is written down a whole step, so the first note G on the manuscript is actually a concert A. For the instrument in Eb, the part would be written down a minor 3rd, distance between C and Eb, so the first note G on the manuscript would be a concert Bb.
If I understand your question correctly, the instrument playing with you is in D and reading the manuscript you provided. Take the three chords G, D, Am, and transpose them up one whole step to A, E, Bm, and consider it A Mixolydian.
All that being said, when it comes down to it, it's what sounds good for what you're trying to do. There are many possibilities of chord combinations to play under that melody. I would recommend trying different ones and see what you like.