Preparing for "differential forms in algebraic topology"?

You would definitely need some basic understanding of manifolds, but I don't think you will need too much. I think you will need definitions, closed submanifolds, immersions/submersions, tangent and cotangent bundles, vector fields and differential forms. Though the book defines differential forms, it chooses algebraic approach. So it would be difficult to read if you don't know what differential forms really are. But you don't need to read the whole book on manifolds. If you will need some extra-stuff, you can always look it up.

Loring Tu's book seems to be a bit too slow (at least for me). I would recommend Frank Warner's "Foundations of Differentiable Manifolds and Lie Groups". You will only need Chapters 1 and 2 (except "Differential ideals"). Altogether it is only about 50 pages, and I think Warner gives a concise and clear introduction to the subject.

In general, I recommend after getting a bit comfortable with manifolds to start reading Bott-Tu. If you will see some unfamiliar term, you can always return back and learn about it. Otherwise you have a risk of spending too much time for learning a lot of things that you don't need for the book, and some of them might not be important at all for you in the near future.

Speaking about exercises in Bott-Tu, there are indeed not too many of them, and most of them are pretty easy. But surprisingly enough, even though I didn't solve tons of exercises, I have learned a lot, and I have gained a lot of skills from this book. So I personally don't think you will need some extra-book with exercises.

But if you will feel that you need some more problems on algebraic topology, there are a lot of nice books. I can recommend Hatcher's book (though it is again a bit too wordy in my opinion), or these notes by Sossinsky. They are not following Bott-Tu book, but there are a lot of common topics.