How small can I expect FDM 3d printers to print? I  was wondering if this printer(daVinci 1.0) had the ability to print very small objects, like insects, coins, or small nuts. (About the size of 1 -2 cubic centimeters)

Here is a link to the printer on the website.

The reason I ask is someone asked me if it was able to, but I have not been able to access the actual 3-D printer for use at this time, just manuals which I have looked through.

So if the 3-D printer was able to print small objects, would a novice be able to do such a thing?

Please let me know if any additional details are needed.
1) If we're talking about FFF/FDM printers:

Accuracy of the electronics and motors allows it, yes.
But how FDM printers work it might be very hard to lay down layers of molten plastic so small as to preserve little details in the X and Y axis, not much of a problem doing 20 micron layer height though (Z resolution).
Check this answer to find out what the X and Y resolution is and what it depends on: https://3dprinting.stackexchange.com/a/509/381

You'll need both a small enough nozzle, as well as somehow cool the plastic because since the printed objects are so tiny the nozzle keeps contact with the surface surrounding it and heats it longer, which might melt the whole object or even char it.

I've seen very few people do tiny prints with success. And the smallest nozzles I know are 250 micron.
Not trying to dscourage you, just letting you know. If it was easy to do I think more people would be doing it and more companies would be advertising their printers as capable of such a thing

So you'll have around 20 micron Z resolution and around maybe 200 in the X/Y.
If that's enough for you, then you could try. Calibrating it all won't be easy, tiniest backlash will be noticeable.

2) It's a lot easier with curing resin 3d printers (SLA or DLP). Most of them actually have trouble printing larger objects, ironically (trouble sticking to the bed and cracking of the 3d print).
Even here badly calibrated lasers would prevent you from doing this and even many Form 1 users have reported their lasers being assembled poorly resulting in poor beam profile.

Size of the laser beam profile (aka laser "spot size") is what determines the X/Y resolution for SLA 3d printers. With the Form 2 its 140 micron, unless you'll get a badly calibrated printer.
For DLP printers it's easier, it's the resolution of the DLP projector divided from the size of the print area.