When I look at the Google homepage in the Inspector and double-click on the end of an HTML tag, I notice that the dotted gray rectangle does sometimes appear, to indicate that I could add an attribute. This is what I want. However, the element also decides that I want to move it: a thick dashed line appears above it, and the whole element moves a bit, exactly as if I were to start dragging and dropping it. Because of this, I can't add attributes to elements, since the text field never appears.

Thanks for reporting this. It is due to the same problem as bug 1210375, which was fixed yesterday and is resolved in the 10-02 Nightly. Please reopen if you can reproduce this one the nightlies go out.

Yep, I can confirm it's fixed as you say. Thanks!