Re: how to replace old registerboxes
IN the past, good installers did all metal duct and did proper sheet metal seaming techniques and/or used mastic. They still didn't use duct tape unless they were hacks.
But yes, you want to use foil tape. You can go to some ****** supply stores and get the better, thicker, wider foil tape. I think foil tape is also sometimes called mastic tape. I would use a zip tie of hte inside, then foil tape, then zip ties the outer sleeve.
What the OP needs are called register boots. They come in different orientations and are sized to the register grill.
1925 Two-Story Stucco Beaux Arts Neoclassical