Thermocouple Equivalent Circuit
You can consider a thermocouple in its usual measurement role as a voltage source with a series resistance. The open-circuit voltage is a nonlinear function of two temperatures, the so-called cold junction(s) and the measuring junction. It can be expressed in terms of a nonlinear function with one parameter.  

The series resistance is a function of the temperature along the wires but usually low enough it can be ignored. You can calculate the loop resistance by the usual formula  R= \$\rho L \over A\$ for the two metals, or simply look it up in a table of thermocouple wire or extension leadwire from the material type, gauge and length. Extension leadwire may be made of a similar alloy to the thermocouple itself in the case of base-metal thermocouples such as J and K (Iron-Constantan and Chromel-Alumel) E and N, obviously that is a much less attractive proposition in the case of types R, S and B, and for different reasons, W, so the thermoelectric characteristics are approximated with different alloys (that are unsuitable for the measuring junction) for the extension leadwire.  

The engineer designing the installation may choose a larger gauge of extension leadwire if the runs are long, in order to keep the resistance relatively low. If everything is within a few meters it usually doesn't matter. 

The resistance is usually &lt; 100 ohms, often closer to 10 ohms. Usually measuring and controlling instruments introduce a small current into the thermocouple to detect a broken sensor, maybe a few hundred nA. Say 200nA and a 20 ohm resistance will cause the sensor to read about 0.1°C high for a type K thermocouple. That offset will increase somewhat with temperature as the resistivity of the metals increases.  

If the thermocouple junction is grounded (which is preferable in many situations) your measuring circuit should accommodate that using a galvanically isolated or differential front end. 

In the old days, thermocouples were made with a defined resistance and fed a meter movement in the pyrometer (with some additional complexity we don't need to go into here), while that may still be true in a few cases, it's rare on industrial instruments. You should just assume the sensor has series resistance 0 &lt; Rtc &lt; 100\$\Omega\$ (in most cases).  



To answer your questions: 


Assume it has Rtc &lt; 100 ohms or whatever makes sense if your situation is not typical. The more current you draw (or bias it 
with) the more the offset or error. How much error is too much? Up
to    you. 
It behaves like a low-value resistor for practical purposes here. 
See 1. the more current you draw or bias, the more offset (and potentially error). Since the resistance varies with the temperature along the leadwire (not just at the measuring junction) any offset will vary with things that are usually beyond your control. 
You can short the thermocouple with no damage. You can even force enough current through it that it glows. As long as the alloys don't
change there is little effect on the accuracy.