Can I do KVL around transistors? (MOSFET and BJT)
I won't say a blanket "yes," since I'm not totally sure what you are asking. Looking at your examples, though:


Yes, if you know \$V_{sd}\$ and \$V_{dg}\$, you can easily calculate \$V_{sg}\$ as you describe. To check the signs, expand each of the terms \$V_{xy}\$ into \$(V_x - V_y)\$. This gives: 
$$V_{sg} = V_s - V_g = (V_s - V_d) + (V_d - V_g) = V_{sd} + V_{dg}$$. I wouldn't call it KVL though (I don't really ever call anything KVL though, so maybe it actually is).
I'm not quite sure what you mean by going "through" the BJT emitter. You can form different voltage loops through the BJT depending on which path you take, and if your problem tells you what the voltage drop is between those two terminals, then you should be able to use those values. The more complicated case is when you aren't given specific voltages and need to use various transistor equations to solve for the voltages in the circuit.