いえ is definitely used informally for いいえ, though it's not super casual. More casual options include いや, ううん, or 違う. That being said, you don't necessarily need a word that means 'no'; you can reply to 「フランスに行ったことある？」with just 「ないよ」
Of course a sentence like 鳥は飛ぶ ("Birds fly.") is 100% grammatically correct. It's 100% standard, too; no slang/dialect is concerned. I'm not an expert, but you never need a professor nor a linguist to judge the correctness of such a simple sentence. Don't be that skeptical about your textbook.
But did your textbook really call this casual style? I ...
I guess it is difficult to give a clear answer to a register question like this, but
つつ can be used in speech;
It is more likely to be used by grownups. At least I don't expect a small kid to use つつ.
Another factor might be a rhythm of the whole sentence or collocations, which I'm not really able to explain.
Two verbs that may be frequently heard with つつ ...