The first plant is celandine poppy. The second is a member of the night shade family datura and both were probably planted on purpose.

It is a little late for the datura to be emerging. Hope some one will post which kind as I like the foliage.

A "weed" is any plant that you do not want growing where you do not want it, so only you can determine if these are "weeds" or not. I have rosa rugosa growing in 3 places where I do not want it growing so therefore that rosa rugosa is a "weed" and I keep cutting it back.

Thanks maifluer! That is certainly a celandine poppy. I looked it up and those are the flowers! I'm so relieved because it is growing everywhere, like a ground cover and if it hadn't been planted on purpose, I would have a lot of work to do!
I'd be willing to bet that the majority of the plants around this place were indeed planted on purpose and I've been making a lot more work for myself than necessary trying to "clean things up".
I'll have to post another photo of one more plant that seems to be everywhere, that I'm not sure was intentional or not.

I just get too much unsolicited advice from people trying to tell me what is a "weed" and what is not!

As Kimmsr stated a weed is a plant where you don't want it. Some only think of smoothly cut grass as the only plant that should be in a garden. Let the seasons roll around for at least a year to see what other surprises you have. If you have the poppy you may have other plants that only show themselves for a couple of weeks in the entire year.

Check (online) your state ag college, plant science department or ag extension service for publications of weeds of your state. You can get a pretty good list and pictures of noxious weeds and other common weeds so that you can identify them, then if what you see isn't in the book, it may be more likely to be a flower that was planted and you can wait to see what it does. By identifying the noxious weeds, you won't be giving any of them a year to create a bigger invasion in your yard if you have them--you can go after them immediately.