Does that mean that I shouldn't give my highest scoring bugs first?
Yes. If you want to get as many items as you can get you should start off weak.
I'm also concerned that if I give lower quality bugs that I'll get lower quality furniture, or will Nat always give you a piece of the Bug set regardless?
How it works is until you meet a certain point level (it was 80 points for me) you will get normal furniture from him every time you break the record. After you break that level of 80? points, each new record after that (including the first) will yield another bug furniture.
So I'm not even entirely sure I can direct the outcome the way I want to but I'd like to try. What should I do to guarantee the most pieces of Bug furniture?
Bugs are graded by three things: rarity (easiest to "manipulate" but you have to find the right bugs), relative length (the game tells you this right when you catch the bug, if you can sort your bugs of the same type by length) and luster/color (this is random and there is no indication of luster until you give it to nat). To maximize the bug items you get, start with not-so-rare bugs and work your way to the rarer ones, working in order of size when you have a bunch of the same bug.
Rarity can roughly be measured by how often you see a bug around. Common butterflies are super common, tiger butterflies are uncommon, emperor butterflies are sort of rare, Raja B butterflies are quite rare, as an example.
Note that the bug off is held monthly, and you probably won't get all bug items in one month. Personally I only managed to get two (I could have gotten more though).
Also not that at the end you'll get a ribbon based on performance. You get a ribbon based on placement, not on how many bugs/furniture you've gotten. A single, high scoring bug will get you a gold ribbon.
The Music Box
The Music Box is actually a family of furniture items, only obtainable by the method you mentioned. Like all other furniture from Cyrus, it is not possible to take one of his products apart into the original materials/furnitures used to make it.
When Cyrus creates the Music Box, the result is a colourful version of the Giant Clam Shell that, whenever opened, will play a little music-box-like version of your song, before closing shut once more.
You can think of the Music Box as a larger, physical, standalone counterpart of the song used to create it, as the Music Box will retain its name when created.
Additional Copies of Music
As for obtaining another copy of the song, if you have access to the Timmy & Tommy Shopping Catalog from upgrading the Nooklings' shop at least once, you'll notice that there is a category in the catalog for music.
Although you cannot order Music Boxes this way, all of the K.K. records you have ever received or will receive in the future will be recorded in the catalog. Each of these records can simply be ordered from the catalog for 3,200 bells.
Therefore, by purchasing your additional copies from the catalog, you are free to request new pieces of music from K.K.Slider each week.
Best Answer
My friend visited my town, requested a song (after I had requested and received a song), and did not receive a copy of the track. So no.