Translation:I think I have a cold.
I agree that 風邪です sounds more formal and polite than 風邪だ, which is the case. However, だ really has the role of declarator, i.e declares a fact.
For example, when you ask a question in casual speech, you wouldn’t say 風邪だ？(Do you have a cold?) because you can’t declare a fact and ask a question at the same time. I got some of this information from Tae Kim’s Guide to Japanese which holds a lot of helpful information.
Hummm.. not quite, I think. です Is always used at the end of a phrase, which is not the case with だ.. besides, the phrase didn't become less formal because there is a "だ" in it. I think it is more useful to think of them as two different constructions, which might have a somewhat close meaning (if it does have a meaning)
I didn't say it is less formal. You just do not use です in front of と when it is used in sentences like と思います or と言います and stuff like that. Of course Japanese is also very flexible with rules when you are talking, for example, with friends, but the normal rule is だ in front of と sentences like this, when it is required, for example by a noun or na-adjective.
"Desu" is the real culprit here. It is peculiar to modern Japanese, or at least it doesn't exist in classical Japanese. Some think it developed from "nite aru" or "nite arimasu." It is pretty much limited to sentence final position. "Da" is the informal affirmative. Both "desu" and "da" roughly mean "is" but be aware that they are not copulas like those in indo-european languages. "To" can be regarded as a particle that indicates some connection to a verb. Among other things it marks the end of a quotation.