You have a minimum current gain given as 1000 in the datasheet, so this means 1mA into the base should result in at least 1A from collector to emitter, assuming the supply can provide this.
The calculation for current into the base (assuming the Arduino pin outputs 5V high) and we take the maximum base-emitter voltage from the datasheet is:
(Vpin - Vbe) / Rbase = (5V - 2.5V) / 1kΩ = 2.5mA;
It looks like your multimeter has quite a high burden voltage (too high a high resistance used for current shunt, so you get a voltage drop across it which affects things) hence the readings being out on the base current and the difference between the supply-collector reading and emitter-ground reading (which should be practically the same - the emitter-ground reading only has the base current added to it, which is tiny compared to Ic)
The darlington transistor has a high saturation voltage (higher than a normal transistor), so a higher voltage supply is preferable for reasonable results, and gain also drops at saturation. In any case, controlling the current in this way is not very practical, since the gain can vary widely between parts, with temperature, etc. Try adding another battery or two, adding a small value collector resistor (capable of the wattage it needs to handle to control max current accurately), and lowering the base resistor to around 200Ω.
If you want to learn about the base current vs gain and saturation relationship, try using a higher supply voltage you know is capable of the current you are testing with, adding a potentiometer (wired as a rheostat) at the base, a small value resistor to give you a known maximum current, and plotting the base current vs collector voltage/current. If you do the calculation you should be able to see how the gain varies and drops approaching saturation. You should get plots similar to the datasheets.
Doing the above in SPICE is also another option if you don't have enough test equipment to make things easy.
From your question, you've only shown 2 data points for your BC546B transistor. If you look at the datasheet, here are the electrical characteristics, specifically the base-emitter on voltage
So the minimum is 0.55, and the maximum is 0.7 to 0.77. The manufacturer's datasheet guarantees your transistor will be somewhere within that range.
Right now, you've only shown us 0.5V and 0.6V for Vbe.
I'd also recommend measuring the base current, Ib.
Best Answer
A NPN transistor basically is a stack of three differently doped areas of a semiconductor. The first is N-doped, the middle P-doped and the last N-doped.
So yes, on the first sight, you can swap collector and emitter, and the transistor might still work. But there's more magic: A real transistor is optimized to fulfill its specs when connected correctly, i.e. the interior is not symmetric. If you swap C and E, you get a new transistor with completely different specs, which you don't know. It's also easy to overload and destroy the transistor by swapping C and E.
An anecdote: When I was about 10, I built my first circuits, and also started to etch PCBs. My first try was an astable multivibrator with two LEDs, and the PCB layout was absolutely mirror symmetric. The circuit worked, but the one LED was ten times longer on than the other. It turned out I had mirrored the second transistor with respect to the first, i.e. accidentally I swapped C and E. This caused the strange behavior, and after removing and soldering the second transistor, the circuit did what it should.