How to figure out the most stable conformation of the diastereomer? Question 1: Which of the following Newman projections represent the most stable confirmation of the diastereomer of the molecule given below?


I guess my main question is looking only at the Newman projection, how does one determine the stereochemistry?
In the molecule to the right, I get for the first carbon, ($\ce{Et, H, Me}$) R (I flipped it after moving the $\ce{H}$ back) and for $\ce{C}$2 ($\ce{Et, Br, Me}$) I get S.


Question 2: Which of the following Newman projections represents a confirmation of the enantiomer of the molecule given below?


In the same vein, for the second question, I thought I should flip all the stereocenters to get the enantiomer since it is a mirror image. But I am not getting the answers the professor provided to us.
To answer your professor's questions, there is no need to determine the R/S configuration of the two carbon atoms. We have to check the clockwise/anticlockwise arrangement of the substituents around the carbon atom, but checking specifically whether it is R or S is unnecessary.
Before moving further, let's have a look at the following image so that we are clear on how to visualise Newman Projections:

Since in the options provided in the question the carbon atom having bromine is at the back, we'll have to look at the molecule in the following manner:

Hence the Newman Projection obtained would be this:

Now, in question 1, option A can be rejected as it's an entirely different molecule. Option 4 has the atoms in the same order as above, so it is the exactly same molecule. In option 2 and 3, the arrangement of atoms is same for the back carbon but different for the front carbon. Therefore, without finding R or S configuration, we can say that options 2 and 3 are the diastereomers!
Out of these two structures, option 3 is more stable as the bulky ethyl groups are at the farthest position from each other.
Moving on to question 2:
You are right in flipping the arrangement about both the carbon atoms and you should get the answer that way. Maybe you are doing some mistake in getting the right Newman projection. Look again at the Newman projection I've provided above and then compare the options with it to see which option has different arrangement of substituents on both the carbons.