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.
I came across a Reddit post that has some research into this problem...
Today was the first day I have seen Katrina in my town (since it may be relevant: I do not time travel). The only other time I've seen and gotten a fortune told from Katrina was once in a friend's town. With this in mind, I decided to see if I could make villagers and then demolish their houses over and over after getting their fortunes read and have it count toward the 20 person total. After having my mayor and my pattern holding character get their fortunes read, I made 17 villagers (same name, face type, and house location). This gave me a total of 19 people getting their fortune read in my town. I went ahead and went into her tent, talked to her, and tried to leave on my mayor character again and she requested the project! So from this, I can conclude that:
-Only players who live in your town can contribute to your total.
-It does not matter if you have your fortune read in your town or another town, it still counts for you.
-You can remake villagers and delete them to contribute to your total.
-Having people visit you does not contribute to your total, instead it actually counts towards your visitor's total 20.
-You can get Katrina to suggest the project on her first visit to your town if you fulfill the requirements that day.
I do not know if the fortune shop will count for sure, but I imagine it would. So you could have someone come to your town each day to get their fortune told and they will eventually unlock the shop in theory. I do not know if Katrina will suggest it to them at the shop or if they will need to wait until she comes to their town, however. I'll do some more testing with this to find out.
TL;DR - If you are paying people to visit your town when Katrina is visiting they should actually be paying you!
So in short, the redditor found that visiting Katrina in any town will increment this count for the town that the player visiting Katrina is from. In other words... if you are visiting Katrina in a friend's town, the counter for your town will increment, but the counter for your friend's town will not.
Best Answer
There are only two things that a villager can interact with: you, and the house.
You
Most of the conversational interaction is just fluff and commenting about your home. The villager will mostly wander around a little until the visit is over.
The House
During the short time frame of the visit, one of these furniture-related events may happen:
When a villager visits your home, speaking to the villager may eventually result in him/her making a remark about your home "missing something". This is usually followed by the villager giving you a present that he/she thinks would go well with your home.
When you visit a villager's home, speaking to the villager may eventually result in him/her offering furniture in his/her home for sale. However, certain items won't be sold to you, such as tables with objects on them, or furniture characteristic to the villager's personality/preferences.
During the end of a villager's visit, you also have to answer a question about how appealing the home is. Despite the fact that the villager can be offended by praise, or even be delighted by criticism, this part doesn't actually do anything meaningful.