I would like to be able to use basic vi commands when the file name change dialog is opened (with cw or cW).
There is no vi-mode in command-line mode, but you can press Ctrl-G to spawn Vim and do the editing there. Or you can use :rename
instead of cw, which will call Vim to edit file list (use nnoremap cw :rename<cr>
if you want to do this always).
At worst I would like to have shortcuts that place the cursor at the beginning of the file name, or delete the file name
Rename by adding keys to perform the change via mappings, e.g.:
" move cursor to the beginning
nnoremap cW cW<c-a>
" delete the file name
nnoremap cW cW<c-u>