I'd choose only the third one, "in any way," for each of your examples. Your second example may be a statement as well as a question. In the right context, "anyway" could mean "in spite of it" in your second question: He tried to convince her to accompany him to the dance, but she argued that her gown was too shabby.
New words, one handy idiom, and a 2-minute quiz — delivered to your inbox to keep your streak alive.