For an example, G3X's normal FR and macro FR are the same according to DPReview. So, what happens when shooting?
Also what does it mean a camera's (FZ2000) normal FR is 30cm? (that means it can focus an object within 30cm-infinity range?) So, it means that it can't focus objects nearer than 30cm?; thus horrible things happen?
If by "focus range" you mean the minimum distance at which focus can be obtained, it isn't a feature of a camera body, it is a limitation of the lens on it. Using better terms, it is called minimum focus distance (MFD).
With a camera with interchangeable lenses (such as a DSLR), using a different lens will yield a different MFR. For example a few Canon lenses :
On a fixed lens camera (such as the G3X), the "macro" mode (for a lens) is usually there to tell the camera that you want to use a reduced focus range (or working range) so the camera will focus faster on close objects. In this case, the range of distances at which the camera can ask the lens to focus on will change from [MFD - infinity] to [MFD - X meters].
On the Canon 70-200, a similar option is possible using a switch, only this time it will restrain the working range from [MFD - infinity] to [X - infinity] (with X > MFD), again to decrease focusing time. As the 70-200 is a "sport" lens and not a "macro" one, its main purpose is to be used on subjects further away.
If your camera has the same "normal FR" and "macro FR" then it means that the "macro mode" will not restrict the focusing range of your camera. It will try first to focus in the ([MFD - X] range (close objects) and if it fails, then it will try the [X - infinity] range.
Usually, no "maximum focus distance" is mentioned by the lens manufacturer because it's infinity (related : What is "infinity focus"?). Given that, you can focus on every object between MFD and infinity. In your case, you are correct when saying that you can in theory focus on everything in the [30cm ; infinity] range.
Below MFD, your camera won't be able to focus (the internal element of your lens moving to adjust focus will reach an extreme position). It will try and fail to focus, stop bothering and the image will just be blurry (sorry, no monster are going to pop-up form another dimension).
In certain case (only for macro application as far as I know), the use of additional element on the lens can reduce "infinity" to a finite value, for example with extension tube (look at What are the biggest differences between Reversal Rings, Extension Tubes and Macro Lenses? and What am I losing when using extension tubes instead of a macro lens?).
It simply means that the range through which the camera will search for focus in standard and macro mode is the same. This almost makes macro mode pointless but it can have other side-effects such as preferring a smaller aperture to increase depth-of-field.
Fixed lens cameras such as the ones you mention measure the focus-distance as the minimum that the camera can focus as measured from the front of the lens. This will usually but not always be when the lens is at its widest focal-length. With the equivalent focal-length (actually angle-of-view given by it) and minimum focus distance, you can figure out the magnification which measures how magnified an object will appear.
By its construction a lens cannot focus on anything closer than its minimum focus distance. It it just no physically possible so trying to focus on something closer results in a blurry image or the camera refusing to take a photo in AF mode. Actual behavior depends on the particular model. AFAIK, Canon is the only manufacturer which makes lenses capable of focusing at 0cm. So it can focus on the dust on the lens! For example, the Canon SX400, SX410, SX420, SX50, SX5w0, SX530, SX540 and SX60 can all do it.
So, yes, something horrible will happen if you try to focus closer, your images will be blurry or not be captured at all. The camera will not break though, it will simply give up focusing. If you are focusing in Manual Focus mode, then turning the focus-ring or moving the focus distance with buttons, on some cameras, will simply stop having any effect.
Note a few differences if you compare this to an Interchangeable Lens Camera sush as a DSLR or a mirrorless: Some lenses have hard stops at the ends of their focus range, the focus ring will simply stop turning. Focus distance of interchangeable camera lens is measured from the sensor, so you cannot compare using that metric. You must look at magnification which is usually not specified for fixed-lens cameras but can be calculated.
