Does it matter whether a thermal expansion tank (for water heaters) is installed above the pipe it tees from (tank's threaded port pointing downwards) or if it's installed below so that it hangs from it (tank's threaded port pointing upwards)?
In general, the tank can be installed vertically above or below the plumbing, or horizontally. Typically, the tank is only required to be supported when installed in the horizontal position. Most smaller tanks are designed to be supported by the plumbing, when installed in the vertical orientation. The most common recommendation seems to be, to install the tank vertically below the plumbing.
You'll want to check the manufacturer's installation instructions, to determine how to properly install the specific tank you're using.
Here's example installation instructions, from a Watts® Potable Hot Water Expansion Tank.
5. Install the expansion tank in the system (refer to Figure 1).
a. The weight of the expansion tank filled with water is supported by the system piping. Therefore, it is important that, where appropriate, the piping has suitable bracing (strapping, hanger, brackets).
b. The expansion tank may be installed vertically (preferred method) or horizontally. Caution: The tank must be properly supported in horizontal applications.
c. This expansion tank, as all expansion tanks, may eventually leak. Do not install without adequate drainage provisions.
My recommendation is that you find a way to support the tank either attaching it to the wall or arrange some hanger support from above or legs support from below. It is generally a very bad idea to hang or set the tank in such way that the connecting pipes support the weight of the tank and the water within it.