The weed seedlings look like hairy bittercress. Cardamine hirsuta. Prolific.

Agree the weed is cardamine hirsuta, and it's more than prolific :) Also known as shot weed from the way it flings/disperses the seeds - don't let it set seed :)

Sorry, I don't recognize the other.

The other plant looks like a speices of Plectranthus, ciliatus?

Could the other be Euphorbia hirta?

Ditto Euphorbia hirta

I believe Cardamine and Euphorbia are spot on, thank you!

Now my question is, are either of them detrimental to the health of the Cordy? I don't actually want to get rid of either purposely, I think they look pretty interesting together. I'd just leave them if I could.

The cardamine will seed itself everywhere. If you enjoy weeding then by all means leave it,otherwise remove it now before it gets to flowering size.
The euphorbia again will not hurt, but i would still remove it.

But if you like the cardamine, maybe you wouldn't mind having it everywhere. Ultimately, it's up to your preference.

The cardamine ended up dying off. I've been keeping the soil too dry for its liking. The cordy is doing great though!