I think that if you are just naming a flower, like rose, peony, hollyhock, etc, then there is no need to capitalize. If it is being used as a person’s name, then you do capitalize. When speaking about a specific rose, like the ones that have a copyright, then at least that part of their name is capitalized, and maybe then the rose part is capitalized, too. An example would the American Beauty Rose, or the Peace Rose, or some roses that are named after people.
I usually don't either. I let them compete for dominance, for the most part. However, if something I particularly like is being crowded out, I might intervene.
A few years ago, a daylily volunteered to grow along our driveway, nearest to the street. It soon grew into the clump of lilies that you see in the front. Given that one of them volunteered to grow here, I decided to plant some others along the driveway, and the following year a couple of others volunteered to grow right next to my lingonberry patch.
You call them daylilies but the ones that grew along both sides of my parents' house we called tiger lilies. They didn't mind if they didn't get full sun and were tolerant of the drip from the roof. I love them. Beautiful picture. It's funny to edit yourself. I said 'both sides' as if my childhood home had only two sides. It had some more, I think.
I did a little research. Papaver rhoeas - also known as common poppy, corn poppy, corn rose, field poppy, Flanders poppy, and red poppy. It's native in Africa, Europe and Asia. It's the poppy from the poem Flanders Field. It grows profusely when the ground is broken up, as in the artillery bombardments in WWI. It was originally only red but selective breeding has resulted in yellow, orange, pink, and white flowers. There is even a black flowering hybrid. I've got the Latin names for the wildflowers I planted. I'll find out more about the rest of them but I doubt any will be quite as interesting as the corn poppy.
I learned something today. A lot of people consider lupines to be the state flower of Maine because they grow wild throughout the state. However, only the blue lupine (Lupinus Perennis) is native to Maine, while all of the other lupine species are considered invasive, and the state eradicates them from state land. Interestingly, Lupinus Perennis is one of seven purple/blue-flowered species of the genus Lupinus that are collectively known as the Bluebonnet, the state flower of Texas. Maine's state flower is, of all things, a pinecone.
We have the wild ones in north Idaho, too, and they only come in kind of a light blue-lavender color. All of the multi-colors don’t grow wild, you have to buy them at the garden store. I have never heard that the other colors were considered invasive, probably because I have only seen them grown in gardens on purpose. This is about how our wild ones look.
Apparently, that was the origin, as it was believed in the 14th century that lupines ravenously exhausted the soil. Current thinking is that lupines simply do well in soil that has already been ravenously exhausted, as it doesn't require rich soil.
Yet another. This has to be close to the last of the new not-seen-before blooms. It sort of looks like a cousin to a dandelion.