Does SQL Server Only Perform Calculations In A SELECT List Once?
Most of the information you will need is going to be in the execution plan ( and the plan XML).

Take this query:

SELECT COUNT(val) As ColA,
       COUNT(val2) As ColB,
       COUNT(val) +  COUNT(val2) As ColC
FROM dbo.TableA;


The execution plan  (opened with sentryone plan explorer) shows what steps it went through:



With the stream aggregate aggregating the values for EXPR1005 &amp; EXPR1006 



If we want to know what these are, we could get the exact info on these expressions from the query plan XML:

&lt;ColumnReference Column="Expr1005" /&gt;
&lt;ScalarOperator ScalarString="COUNT([Database].[dbo].[TableA].[val])"&gt;
&lt;Aggregate AggType="COUNT_BIG" Distinct="false"&gt;


With the first compute scalar computing ColA &amp; ColB:



And the last compute scalar being a simple addition:



This is reading it as the data flows, in theory you should be reading it from left to right if going over the logical execution. 

In that case, EXPR1004  is calling the other expressions, EXPR1002 &amp; EXPR1003. In turn these are calling EXPR1005 &amp; EXPR1006.


  Would CalculationA and CalculationB, each be calculated twice? Or
  would the optimizer be clever enough to calculate them once and use
  the result twice?


Previous tests show that in this case ColC is simplified as an addition of the calculations that are defined as ColA &amp; ColB. 

As a result, ColA &amp; ColB are only calculated once.



Grouping by 200 distinct values

If we are grouping by 200 distinct values (val3) the same is shown:

SET STATISTICS IO, TIME ON;
SELECT SUM(val) As ColA,
       SUM(val2) As ColB,
       SUM(val) +  SUM(val2) As ColC
FROM dbo.TableA
GROUP BY val3;


Aggregating down to these 200 distinct values in val3



performing the sums on val &amp; val2 and then adding them together for ColC:



Even if we are grouping on all but one non-unique value, the same addition should be seen for the compute scalar.



Adding a function to  ColA &amp; ColB

Even if we change the query to this:

SET STATISTICS IO, TIME ON;
SELECT ABS(SUM(val)) As ColA,
       ABS(SUM(val2)) As ColB,
       SUM(val) +  SUM(val2) As ColC
FROM dbo.TableA


The aggregations are still not going to be calculated twice, we are simply adding the ABS() function to the resultset of the aggregation, wich is one row:





Ofcourse, running SUM(ABS(ColA) &amp; SUM(ABS(ColB)) will make the optimizer unable to use the same expression for calculating ColC.



If you want to go deeper in when this happens I would nudge you towards Query Optimizer Deep Dive - Part 1 (until Part 4) by Paul White.

Anther way to dive deeper into query execution phases is by adding these hints:

OPTION 
(
    RECOMPILE, 
    QUERYTRACEON 3604,
    QUERYTRACEON 8605
);


This will expose the input tree as created by the optimizer.

The adding of the two previous calculated values to get ColC is then translated to:

AncOp_PrjEl COL: Expr1004 

ScaOp_Arithmetic x_aopAdd

    ScaOp_Identifier COL: Expr1002 

    ScaOp_Identifier COL: Expr1003 


This information is already present in the Input Tree, even before the simplification phase has taken place, showing that the optimizer immediately knows that it does not have to perform the same calculation twice.