Echoing gollyzilla a bit, a 2 pane display would be great with different directories in each. Drag & drop would be nice, but action buttons (copy, move, rename, etc) would be OK. If the buttons were user assignable, that'd be even better. Something like Directory Opus or X-tree.
Multiselect is essential. I don't know how that would work with drag & drop so the action buttons might be necessary.
Archive support would be great with integrating your archive app. 2 pane helps here too. Archive file in one pane, destination in other pane. Option to extract to that directory or create new container directory. Reverse would be nice too - select multiple files & create archive.
About second tap funstionality. First idea - a tap & hold pop-up menu with copy, move, rename, delete, share, add / remove from archive, open, open with, etc. Could also use tap & hold on empty space (or action buttons) to do things like create folder or file.
Second idea - first tap selects. If second tap is on name, rename. If second tap is on icon, do other action.
How do you think you'll implement multi-select? Check boxes, button to enter select mode, some kind of multi-tap ala meta-tap? Something else?
A nice feature for multi-select would be multi-rename.
Hope some of this is useful.