Which correlation formula should be used when we add up many measurements of the ordinal type? Let's say, I have two set of questions; one set is to detect depression (var1) where the response of each question is in Likert scale and another set of questions to detect hopelessness (var2) where responses are also in Likert scale. Assume that, I have 50 samples who will be tested for depression and hopelessness. Now. I have to find the correlation between the variables, var1 and var2. 
To do that, I want to calculate the depression or hopelessness score for each sample by summing the questions score (i.e. Likert Scale value for each question) of each questionnaire (depression or hopelessness). Then, find the correlation using formula.
What does create problem?
As each question of our questionnaires is in Likert scale, they are ordinal (Likert scale is ordinal). 
BISHOP and Herron remarks 

If  the  data  are  ordinal,  then  non-parametric statistics are
  typically considered  the  most  appropriate  option  for analysis.

This one also says

For ordinal scales, the correlation coefficient which is usually
  calculated is Spearman’s rho.

However, this one in an answer of SE question, says

But when you add up many measurements of the ordinal type, which is
  what you have in your case, you end up with a measurement which is
  really neither ordinal nor interval, and is difficult to interpret.

On the other hand, Muayyad Ahmad remarks as an answer of a question in RG

For Likert scale, the items are ordinal, but usually we do summing for
  the items to get total score, which is considered as continuous
  variable. then we do Pearson correlation

David L Morgan also remarks there

Twenty-five items are a lot for a single scale, which is what you
  would get if you simply added all the items together. That is standard
  practice for creating a continuous variable from a series of ordinal
  Likert items, but it requires careful attention to whether thee items
  do indeed form a single scale.

Therefore, as we are adding many measurements of the ordinal type questions score in case each of the questionnaire (depression and hopelessness scale), will the value (summation of all questions score for each person and questionnaire) still remain in ordinal type? If it becomes like an interval variable and continuous, can you please refer a book or paper which says this one?
 A: Have you tried plotting your data?  Both histograms of the 2 summed scores and a scatter-plot of them against each other.
When you start summing up responses (even if they are ordinal to begin with) the Central Limit Theorem comes into play and if you sum enough things together that are not too close to boundaries, then you will start to see something that is approximately normal.  But if many of your answers are at one or the other of the possible extremes or the individual responses are highly correlated, then you may not have enough for the CLT to help yet.  You need to do some data exploration to get a feel for this.
In your case there is not one perfect/correct answer, but there could be many approaches that are good enough, but which are good enough will depend on things that you will see in  the plots (and combined with your knowledge of the science behind the data).  
If the histograms look fairly bell-shaped and the scatter-plot looks pretty linear then you will not see much difference between Pearson, Spearman, and other correlations and it is probably fine to use any of them.  But if there are clear clusters on the scatter-plot and/or U-shapes in the histograms then any single measure of correlation will probably over-simplify the relationship and distract more than help understanding.
I am a big fan of simulation for helping me understanding what is happening and what may happen.  You could randomly generate some data under different conditions (uniform across the possible answers, mostly low, mostly high, mostly low or high unlikely in the middle, etc.) and compute the different correlations and plots and compare them under the different situations to see which give you the most meaning compared to the simulation conditions.  
