How to delete (not cut) in Vim?

The black hole register "_ will do the trick, but there is a better solution:

When you enter the line back with the p command you are pasting the contents of the (volatile) default register "", which has been overwritten by dd. But you still can paste from the (non volatile) yank register "0, which won't be overwritten by the delete command dd.

So these are the commands you want to use as per your example:

yy
dd
"0p

Use the "black hole register", "_ to really delete something: "_d.
Use "_dP to paste something and keep it available for further pasting.

For the second question, you could use <C-o>dw. <C-o> is used to execute a normal command without leaving the insert mode.

You can setup your own mappings to save typing, of course. I have these:

nnoremap <leader>d "_d
xnoremap <leader>d "_d
xnoremap <leader>p "_dP