The story about buttercups giving butter its yellow colour is certainly false, because it is poisonous to cows and other farm animals, and also to humans.
Presumably it is also poisonous to domestic pets. However its toxicity is not usually a serious problem, because its taste is so bad that nothing will eat it unless there is no alternative food available!
It spreads mainly by sending out runners which root and form new plants. Your garden won't be overrun by buttercup seeds germinating everywhere, but it will spread quite fast, and you don't want it in your lawn for example. It dies down in early summer and the tuberous roots are dormant in the soil till the following spring, so once it has finished flowering it "disappears" and is hard to eradiate.
The "wild" species in your pictures really belongs in a hay meadow, not a garden. There are some cultivated varieties sold as flowering plants, see https://www.rhs.org.uk/plants/search-results?form-mode=true&context=l%3Den%26q%3DRanunculus%26sl%3DplantForm&query=Ranunculus for example.