<< Firstly: I have two existing sata 1 HDD's in the old pc,(both contain data but are non-os drives), can I simply unplug and plug into the new pc with sata 2 sockets? >>
Yes, you can. They're backward compatible.
<< Second: If so, will I need to wipe these drives first? Lot of data on them so I hope I don't have too! >>
No, you don't have to, just plug & play.
<< Third: The new desktop is an i3, running W7 Pro 64 bit, 4G memory, with dual onboard display. Can I run the 2 monitors via the board without any problems? Or should I add memory or add a dual display graphics card? >>
That'll depend on what dual onboard display the motherboard has, dual monitors should run using the D-sub (VGA) and either DVI-D or HDMI, the DVI and HDMI can not both display at the same time. Or you can use DVI-D or HDMI with display port. If not you might have to add a dual display graphic card.
<< Lastly: I plan to use my existing monitor(s) Acer AL1916w, will these simply plug in and work? >>
Yes but you might have to get VGA to DVI/HDMI adapter if either the onboard or dedicated graphic card doesn't have VGA connection.