Can an LED driver operate within the whole range between min and max specs effectively, or is a load right at the max specs too large?
As Eric Gunnerson pointed out, a LED driver is a kind of switching power supply.
Switching power supplies are notorious for taking much longer to get working than anyone expected.

People keep Underestimating the Complexity of Power Supply Design.


  Can an LED driver operate within the whole range between min and max
  specs effectively, or is a load right at the max specs too large?


It may.
I prefer to design systems such that every component is well within its specified limits,
rather than dancing right on the edge of the cliff.

In particular, the "Absolute Maximum" ratings are useless for designing new systems -- parts are not guaranteed to operate properly just below those absolute maximums, they are merely guaranteed not to be permanantly damaged.
The devices are guaranteed to operate properly only after conditions have returned to the normal operating range.


  What is the difference between an N-channel MosFET controller and a
  driver?


A LED driver is the stuff that goes between some power supply (often 12 VDC) and the LEDs.
A LED driver is usually composed of a PCB with various components soldered to it -- connectors, inductors, capacitors, and a controller chip.

As the power flows from the power supply to the LED, LED drivers invariably use some kind of MOSFET to control that power.
Some LED drivers use a controller chip with a built-in internal MOSFET.
Other LED drivers have a controller chip that is designed not to handle the power directly (the LEDs do not directly connect to the chip), but instead the chip turns on and off an external MOSFET that is connected to the LEDs.
The vast majority of such LED drivers use "N-channel MOSFETs", although a few use "P-channel MOSFETs".

The LM3421 and LM3423 devices are such a controller chip, designed to be placed on a PCB and connected up to external N-channel MOSFETs, inductors, etc., in order to build a LED driver.


  Iout(max)(A): 5
  
  where did they fish the 5A max current from, if it
  says it can only do 350mA?


Circuits have many parts. Different parts often have different amounts of electrical charge flowing going through them.

The current going through the LM3421 chip itself (if the circuit is properly designed) is always relatively small.

With a LED driver controlled by the LM3421 chip, if the other components (resistors, inductors, nFETs, etc.) are selected appropriately, the current going through the LEDs can easily reach 5 A, as explained on p. 11 of the datasheet.


  These specs I do not know the meaning of:
  
  Vref(V): ... Iq(typ)(mA): ... Type: ... Dimming method: ...


It appears that you are reading a web page written by a well-meaning web designer ( http://www.ti.com/product/lm3421 ) rather than a data sheet written by the experts on the part ( http://www.ti.com/lit/ds/snvs574e/snvs574e.pdf ).

While this web page looks accurate as far as it goes, and while this 60 page datasheet may appear to be more difficult to understand than a much shorter web page, please believe me when I tell you that the reason the web page is much shorter is because it leaves out a bunch of crucial information that you absolutely need to know in order to wire up this chip and get it working properly.

And since you need to read the datasheet anyway, there's no point in even looking at that web page, since it's not going to tell you anything that isn't in the datasheet.


  Vref(V): 1.24


The reference voltage is 1.24 V. (Or is it really 1.235 V?).
You need to know this voltage, and the rated current of your LEDs,
in order to pick the right resistor (Rcsh) to wire to the CSH pin.


  Iq(typ)(mA): 2


The quiescent current (Iq) is 2 mA.
That's the current this chip continuously drains from the battery when the LEDs are turned off and it's doing nothing, aka "vampire power".

If you want something to run for a long time off a CR2032 battery,
you'll want to put this chip back and use some other LED driver with lower Iq.


  Type: Inductive


This chip uses an inductor, as opposed to linear current regulators like the LM317 or charge pump converters.


  Topology: Boost, Buck-Boost, Sepic


Many switching voltage regulator chips are designed to be a part of a buck regulator, and it's awkward to get them to do anything else.
This chip is more flexible, and can fairly easily be wired up in any one of these 3 types of switching voltage regulator.


  Adjustable Switch Frequency


As explained on p. 12 of the datasheet, you can adjust the switch frequency by adjusting the resistor and capacitor that you connect to the RCT pin.


  Enable/Shutdown


This chip has an EN (enable) pin.


  Dimming method: PWM (although I am not sure how that is controlled)


This chip can respond to either an analog signal or a digital PWM signal
to adjust the brightness of the LEDs.

Page 1 of the datasheet shows that a PWM signal (perhaps from an Arduino)
applied to pin 8 (nDIM) controls the dimming.