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How do we know whether a formula needs g or kg? some formulas assume g while others assume kg. What is a good rule of thumb? Is it normally that mechanics equations such as 1/2mv used kg whereas other equations in quantum mechanics use grams?
you would normally use grams unless the question uses or states otherwise or if the calculated number is very large, making it more convenient or logical to use kg. Typically, you'll be able to tell what is appropriate.
Unless otherwise specified, the general rule of thumb is to give the answer in the units the givens were presented in. It is also good practice to recognize if your answer contains a 10^ x that better corresponds to another unit (ex: picometers 10^-12, megameters 10^9, decimeters 10^-1, etc.), but always relay back to the general rule of thumb.
Unless the formula explicitly states that the formula requires the variables to be inputted in terms of certain units, it does not matter which units you use in the formula. What you should do is to keep track of the units and convert the units to another units using conversion factors whenever you find it is appropriate to do so. Based on the 1/2mv2 term that is being asked, it seems that the final answer you may be looking for is in terms of Joules. Just remember that 1 Joule = 1 kg*m2/s2. This should help you understand that even if you start with g, you can convert g to kg then convert to J if needed.
It was recommended by my previous TA to just go off of what the problem begins with unless stated otherwise. For instance, if they give you grams for any part of the problem, your result should reflect grams.
As stated above, it is important to always follow the units that the problem gives you. It is also important to note that certain units such as joules are given in (kg(m^2))/s^2 and that the problem will not work if you use grams in it.
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