Desktop of course. Is it even feasible to trade on a mobile device? The screen is too small there's no way you can read info (orderbook, charts, volume, etc) properly. Even with a 14-inch laptop I often wish my laptop had a bigger screen.
The platform doesn't matter much to safety I think. Just make sure you set up your independent 2FA properly and you should be safe.
Yeah, it is very feasible to trade on a mobile device but that would depend on your pattern of trading as a day trader obviously cannot easily be able to do anything on a mobile device, but for short term or long term trader, the likes of Tabtrader application where you can make use of the exchange API on the application, and also be able to do analysis with the indicators present, can make it easy for you to make trade decisions on the go without having to go on a desktop or login to your exchange every single time.
Nevertheless, when it comes to safety and security, that has much to do with the exchange and the user itself (like you said with the 2FA) and for now, for exchanges that I am personally convenient with; bittrex, binance and a little bit of KUcoin is doing it for me.