Slow down a bit - what is the harmonic structure of a tune, & how would I find out what it is?

Put very simply, most conventional Western tunes have a melody line which rises and falls - goes where it wants to. Underneath that melody, you can have a chord sequence which supports the melody and adds harmony. For conventional written music, the composer will have written the chord sequence or the underlying harmonies - and, of course, you can do away with them if the fancy takes you and use your own.

So, taking your Three Blind Mice as an example. The first 12 notes of the tune, in the key of C, go roughly:

E D C - | E D C - | G F E - | G F E - |

The chords which conventionally underpin that tune are:

C G7 C C | C G7 C C | C G7 C C | C G7 C C |

So, taking the top melody line - shown again but in italics - and putting the chord notes for it vertically below, we get the harmonies:

E D C - | E D C - | G F E - | G F E - |
--------------------------------------------
C F C C | C F C C | C F C C | C F C C |
G D G G | G D G G | G D G G | G D G G |
E B E E | E B E E | E B E E | E B E E |
C G C C | C G C C | C G C C | C G C C |

The chord structure below the melody can be called the tune's harmonic structure. This is a very simplistic example. My main point was that all tunes can have several possible underlying harmonic structures or chord sequences - whether you choose to use them or not is your business. Parallel 5ths are in themselves a structure, but 2 notes on their own do not a chord make... :-)