Well, Whitney's condition A is implied by condition B. A good reference is the book of Banagl, Topology of stratified spaces, Chapter 6 if I recall it correctly.

Do you know any example?

Besides that, the methods described above can be used also in different settings: For example, let $N$ be an algebraic group, with $N_0=T$ is a torus. If $N$ acts on some variety, such that the action restricted to $N_e$ makes $X$ to a toric variety, then you can proof by similar methods that $D^b_N(X)$ is equivariantly formal. (Note that formality of $D_{N_e}(X)$ was shown already by Lunts)

Yes, you are right, and this being "careful" is the whole point of the diploma thesis mentioned above. If you really need this thesis I can scan it and send it to you via eMail.