You earn medals by participating in Tours on Tortimer's Island. You can either do this solo, with friends, or through random matchmaking through the Internet.
The number of medals you are awarded is dependent on how well you do in each minigame. The better your performance, the more medals are rewarded. There are also 3 difficulties, from 1-3 stars, and the higher the difficulty, the more medals you earn.
For example, during the Easy Hammer Tour (1 star), I hit the machine 91 times, which gave me a Gold Tortimer award (these are bronze/silver/gold and aren't tangible, just an award to judge how well you did) and 6 medals. During the Level-Up Hammer Tour (2 stars), I hit the machine 80 times, which gave me a Gold Tortimer award and 8 medals.
In addition, when playing with multiple people, you earn Bonus Medals based on how well you did in comparison to them. The way the games work is that they are a mix of cooperative and competitive. You all have to do something that works towards a collective goal, but you do get bonuses based on how well you did. So, again using the Hammer Tour as an example, you have to hit the machine 30 times within 3 minutes. Each person's hit would go towards the total, but at the end, they would also get bonus medals for how many times they hit it.
Once you earn 50 medals, you can earn membership to "Club Tortimer", which allows you to participate in Tours with random people on the Internet. If you're not a member, you can only go on Tours with people with whom you are friends. This is handy if you don't have friends that you can play with or they're on at times when you're not available, etc.
In conclusion, the fastest way to earn medals is by playing with multiple people as opposed to solo, and playing on higher rather than lower difficulties (and performing well, to boot).
There is not a bug cost limit, there is a bug spawn limit. From my own observations and from discussions on the Animal Crossing subreddit, the Island can only spawn a maximum of 5-6 bugs at once.
If most/all of the bugs you currently have spawned are "trash" bugs, like Fruit Beetles and Wharf Roaches, scare them away so the higher value beetles can spawn. I'm disinclined to believe that there's a "worth" limit, given that I've had Golden Stags spawn with Horned Hercules and Goliath Beetles at once before, and I've never seen any other discussion implying that there is any sort of invisible worth ceiling.
Remember that spawns are still random, so in doing a circuit around your Island, you can still end up seeing only 1-2 bugs just because of chance. Be diligent about chasing away the low value bugs (especially the Wharf Roaches, as more than one tend to spawn at once), and you should end up seeing more high value bugs.
Best Answer
Gyroids appear after rainy or snowy days replacing fossils that would spawn. It is not immediately after the rain stops, but the very next day. So you basically need to wait until tomorrow after 6 AM.