There do not seem to be hall effect current senors available for small currents, say on the order of 500mA. I'm guessing this is due to some technical or physical limitation. What is it?
Electronic – Why don’t hall effect current sensors exist for low currents
current measurementhall-effect
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It should make no difference as the sensor just detects the magnetic field emanating from the current-carrying conductor, which is just a piece of metal. It may act like an inductor, however if the sense wire is coiled up for higher sensitivity.
The only thing that I can think of is you may have had a bad connection somewhere that caused the drop as you should be seeing the same thing on both sides of the load.
4.5V to 76V is the Input Common-Mode Range of that MAX4080. What that means is that both of its inputs (their average) can get that high above ground without damage to the amp. (Furthermore, for this apm, its common range is independent of the actual supply voltage; this is explained on p.9 of the datasheet.) Also, this range is not about the max difference between the inputs. The latter is
Differential Input Voltage (VRS+ - VRS-)..............................±80V
So yes, you can use it pretty safely in this application, as you can't really get the voltage drop (on any resistor) to exceed the supply voltage; even when considering the supply ripple in this case.
Do note however that those input signals cannot exceed the (power supply) rails of the amp for actual measurement purposes, and depending on the amp's input design the limit can be much lower; the absolute values in the datasheet are for damage prevention.
What you actually care about for measurement is the full scale sense voltage, which actually depends on the sub-model of that amp as follows:
(The footnote says that "Negative VSENSE applies to MAX4081 only" since the 4080 is unidirectional.) Also this scale corresponds simply to the built-in gain as explained later:
Total gain = 5V/V for MAX4080F, 20V/V for the MAX4080T, and 60V/V for the MAX4080S.
Unlike an opamp, you cannot change the gain of this 4080 IC; the feedback is internal. You can of course add another amplifier stage if [somehow] needed.
So you choose your sense resistor accordingly. For example with the 4080F, the max drop it can measure is 1V, so for 8A max that means the max resistor you could use is 0.125 ohms.
They even give you a table with pre-calculated values:
You'll also want to read the parts of the datasheet about the OUT High Voltage and low respectively, to know how much the ouptut of the amp can swing. These depend in part on the supply voltage, can get no closer than 0.27V from the rail. So to actually get those 6V output, you need a supply of at least 6.27V for the amp... which in your case is easily achieved (you have 15V).
Regarding 2nd question, Hall sensors aren't usually used in this application. And if you wonder why I'm not going into more detail on this: the usual rule here is one question per question-post.
Best Answer
Hall effect current sensors measure the magnetic flux generated around a conductor carrying current. As such, the sensitivity is limited by the noise floor due to extraneous magnetic "noise" in the vicinity of the conductor.
This can be overcome to varying degrees by concentrating the magnetic flux due to the current carrying conductor, by a fairly simple means: Pass the current to be measured through a coil surrounding the hall effect sensor.
For example, section 12.1 of the Melexis MLX91206 linear hall effect current sensor datasheet illustrates the use of a coil for measuring small currents:
In practice, so long as the design can tolerate an inductance in the current path, the MLX91206 works well enough down to 100 mA current for full-scale output. When measuring supply rail current, this can actually be leveraged to added advantage by using the inductance for ripple suppression, "for free".
Conjecture: It might be worth exploring whether a non-rectangular (toroidal) coil provides better extraneous magnetic noise attenuation than the rectangular form - perhaps even lower currents can then be measured.