I cannot answer your first question, but it oddly sounds a bit like an
English/Scottish/Irish?  folk tune at the very beginning, but my guess
is that observation is unlikely to have anything to do with the tune.
It is quite fascinating.  Obviously tonal, obviously regular phrase
structure, with an equal number of measures per phrase, not unlike a
folk tune... The harmonies used at the end of third phrase and beginning
of the last phrase have a sort of Brahmsian sound to them.  The accompaniment
figurations are suggestive of the Romantic period.  As to when it was
written, I find that almost impossible to answer these days.  One can
find "new age" music written today that will sound similar, but not
usually with that independence in the lines in the accompaniment...nor
that variety in the harmony.

The second "interpretation" obscures the tune by bringing out what I
would consider the subordinate lines. I like both.

Is melody important...it depends on how one defines melody...for me there
is music that is more gesture than melody.  I think of a composer like
Paulus.  Some of his music strikes me as series of gestures strung
together, yet, at his best, the progression pulls me along.

The tune you presented, with that accompaniment, sounded rather nostalgic
to me, a thoughful look back.  The melody speaks to me, but melody alone
rarely "does it" for me.  For me, a melody has its greatest significance
within a context of the form and its setting.  Why do I like a theme?
It is easier for me take themes I like and then look for similarities.
Most of the themes I like have often make use of some ambiguity between
major and minor, will often feature the intervals of the fifth and fourth,
make use of hemiola, etc.  The most important quality is that the theme
set one up for an expectation and then not take you there, but take you
somewhere better than you could have imagined.  This theme and the harmony
held few surprises, except towards the end of that third phrase through
the fourth phrase...a hint of Brahms and a hint of Schumann, where it
became much less like a folk tune and then left me at the end wondering
where on earth it was going to go...

I can find great melodies from composers not necessarily known for
their melodies, but it comes from the context.  For example, when the
B theme of the last movement of Hindemith Symphonic Dances returns, I
find it very inspiring and greatly moving.  When the short introduction
in the first movement of the Prokofiev Second Piano Concerto returns
in augmentation, just after the cadenza...wow.  In the last movement of
the Bloch Concerto Grosso No.1, fugue...when the theme returns in triple
augmentation...another wow...the B theme of the Rondo finale of the
Piston Seventh Symphony...Barber's song, Sure on this shining night, the
main theme from the slow movement of his piano concerto, especially in
the iteration with the rapid accompanying figures in the right hand...the
main theme of the slow movement of the Ravel Piano Concerto, the cadenza
of the Ravel Left Hand Concerto, take away the accompanying figurations
and it is an ok theme, but with them...another wow...the aria "Lord Jesus
Christ" from Barber's Prayers of Kierkegaard, the horn solo towards the
end of the Firebird, etc.  But I find it almost impossible to take that
horn solo out of the context of material which precedes it and its
accompaniment.

Then there is the 18th Variation in Rachmaninoff's Paganini Rhapsody,
a simple inversion of Paganini's theme. How about the opening theme of
the Hindemith Second Cello Concerto, the main theme used by Walton for his
Hindemith Variations...La Follia...the Dies Irae...while it is a wonderful
theme, for me what Liszt did with it in his Totentanz makes it a great
theme.

Perhaps we could list favorite fugue subjects: c minor BKI well tempered
of Bach; Harris Third Symphony; the aforementioned theme from the Bloch;
etc.  Then there is that wonderful fugue in the Hindemith Concert Music
for strings and brass, not necessarily a great theme, but a great fugue.

Doing this exercise I wondered if you were asking if an easily recognizable
theme was important.  I would suppose that could be a notion that suggests
that the role of a theme is to provide an aid in figuring out the form
of a work. So is the question, does knowing the form, or having recognizable
elements which return in a piece, a good thing.

For me, not having any reoccuring theme, gesture or idea, makes for
difficult listening, that is why I find a work like La Mer to be such
an amazing work.  To the best of my knowledge, it is rather without a
clear formal structure other than its dramatic content.  Yet, it carries
us along.  It was rough seas for some of the first audiences, but yet
it quickly gained approval.  I find it difficult to think of works that
enjoy such popularity that do not have some cyclic aspect to their
structure.

I believe that thinking of music purely in terms of a tune can be limiting
to both composer and listener.  I am reminded of a story my friend Kent
Kennan told me.  Kent was in Rome the same time Barber was.  One evening
Barber played and sang Dover Beach for Kent.  Kent in turn played his
Night Soliloquy for Barber.  Kent told me Barber was very critical of
the Night Soliloquy..."no consequent phrase!" Oddly enough Dover Beach
is one of Barber's least often performed works and the Night Soliloquy
was Kent's most successful piece.

Sorry for rambling, but your example and question reminded me of some
of the basic things I used to bring to my listening...and that maybe I
should not neglect them as much as I have.

Karl