I've never had a blue rose and I've caught at least two. The Mantis isn't very common so you just have to be patient

I don't know the exact number, but the more you use an item the gold version will be available in the store. Gold versions are 1000 coins each

It says you have to press down on it for a few seconds. It doesn't always work well for me, but keep trying and just hold your finger on the item you want to move.

Wait for it to spawn in your town

You plant it and wait for it to grow. Then you pick it after it has grown and go back to the museum. He'll see it in your inventory and take you to a place where you have to dig something out of the ground. Make sure you have a shovel!