What platform? Download the right version, which one is documented here:
No, since Monero is not yet available from Debian package repository, the only way is to install it from the Monero website. Apart from these hints, installing monero in Whonix does not differ from installing monero on any Linux based distribution.
To my knowledge, there are no gpg signatures for monero yet, so best you can do is verify the SHA has. So it’s either use it as per conditions set by monero developers or not use it. (Whonix does not do anything special here.) Update: gpg verification instructions here: https://web.getmonero.org/resources/user-guides/verification-allos-advanced.html
Having said this, you can ignore the popup. Starting it from dolphin will not work anyhow, at time of writing and to my knowledge there is no gui for monero yet. So you have to start the command line version from command line anyhow. Update: There now is also a GUI version.