A supply of 10 volts across 1k means 10mA is the maximum current you can get thru the MOSFET. That is the problem with your experiment - no matter how much more you turned the juice up on the gate, the aiming point for drain current is 10mA and can not be exceded no matter how hard you try.
Of course Vds decreased - the mosfet was turning really hard-on and acting as a very small value resistor. This small resistance forms a potential divider with the 1k to make a very small voltage that is nowhere near 10V.
The Source and Bulk do not have to be connected.
In power devices, and especially in discrete transistors, the S & B are built very close together and shorted. This improves the breakdown voltage performance of the transistor.
In an IC, in some CMOS processes, the B of NMOS devices is always substrate (ground), and so in structures such as NOR gates which have 2 NMOS in series, the 2nd NMOS doesn't have the S=B.
Generally performance (gm, current) is better with S=B, but some technologies don't allow the B to be separated from the substrate for NMOS devices. PMOS devices in an IC generally can have separate S & B connections.
If you connected the B of PMOS to GND, you would have a parasitic diode from S to B (GND), and so your supply would be shorted (unless you wanted to run on a very low supply voltage of << 0.6 V). Some very low voltage circuits do use this technique.
A MOSFET connected as a diode will generally have worse performance than a PN junction in terms of the 'sharpness' of the curve. However, FETs with low threshold voltage (say 0.4 V or lower) will turn on at a lower voltage than a diode will, and this can be useful in low voltage circuits. For the same reason that the B is always substrate in some CMOS ICs, there isn't the flexibility to use a PN junction as a diode in all circuit configurations. If the PN junction of a PMOS is used (P = Source, N = Bulk), then there are some additional parasitics that need to be considered that make this not useful generally.
Best Answer
It acts as what it is: the body diode in parallel with the MOSFET source-drain resistance.
Current is shared between the two until Vgs is high enough that the voltage across the source-drain resistance falls below the forward bias voltage required to allow the diode to conduct. At this point, the source-drain shorts out the body diode and all current flows through the source-drain, rather than the body diode, and has a resistive-only voltage drop associated with it.