I've got an A1200 and a Iiyama monitor, and have experience of Power Computings scan doubler, so reckon I can answer your questions :-D
For a start, you CAN connect the Iiyama to an A1200 without a scan doubler. There are 2 problems with this though:
a) You need a converter to change the A1200 video connector socket to an SVGA one which the Iiyama (and most other standard PC monitors use). I think I've seen these on sale for about 10UKP
b) You need to change the screen mode of AmigaOS to use a higher scan frequency than PAL or NTSD. Double PAL or Double NTSD will work, and if your particular Iiyama monitor is a multisync one, some other modes may also work. However, this will only work for games and applications which work with AmigOS. Most games and some applications bypass AmigaOS, and such software in general will not work as they default to PAL or NTSD. You will also not have any access to the Amiga early startup screen.
If you buy the scan doubler, it will double scan PAL (and I believe NTSD) screens from the Amiga to be compatible, and so almost every game and application available will work with the monitor. For modes other than PAL or NTSD, those are simply passed through the scan doubler. If they are then of a compatible frequency, they will be displayed, if not the Iiyama will give a "frequency out of range" warning. In addition, with a scan doubler you won't need any adaptor to convert the Amigas video port to the connector on the monitor as te scan doubler deals with that. Finally, it will also allow you to see the early startup menu
I've personally actually bought the Power computings internal Scan doubler with flicker fixer. This fits totally internally to the A1200 and has an SVGA monitor connector which can hang out or a blanking plate at the back of the A1200. It's great as it also leaves the generic A1200 video port free (ideal if you use a genlock). All games and apps work 100% on my Iiyama Vision monitor with no trouble. The flicker fixer part means that the output is de-interlaced so you don't get any dreaded flicker when using PAL interlaced screens.