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The Macmillan Dictionary regard touching as a formal preposition, and other dictionaries nearly related the prepositions concerning, about, touching as synonyms. So is it fine to use all of three prepositions in written exams, as in:

1. They had discussed touching the beauty of that place.




2. They had discussed about the beauty of that place.




3. They had discussed concerning the beauty of that place.

I don't have an idea which one is more preferable in written exams or other formal speaking occasions?

And the following usage of touching as a preposition lead me towards the confusion, as it looks like present progressive tense:

• It is touching them.

In this case, shouldn't it be like this:

• It is about them.

Notice how the example usage in Macmillan is not a complete sentence, and is preceded by a noun:

questions touching party politics or religion

whereas your example uses a verb before the word touching:

They had discussed touching the beauty of that place.

Therefore, a better example usage would be:

They had a discussion touching the beauty of that place.

Although, as another answer mentions, this isn't a common usage in contemporary English, and it is prone to sound a little awkward even when used grammatically.

One more common, contemporary phrasal verb is touch on. Macmillan says:

touch on something to mention something when you are talking or writing

So, one could say:

Their discussion touched on the beauty of that place.

The use of touching in a context where it means something similar to about or concerning is, as @jason-bassford pointed out in a comment "above" [currently], usually in the form of touching on, and implies that [the topic] is dealt with in passing rather than being the central theme of the piece in question.

Using about or concerning implies that the subject is the central concern [sic] of the work, and the latter can imply concern in the sense of worry: for example I sent you a letter concerning [X] would suggest you should be concerned about [X].

Examples:

1. 
   Henry IV, part one is about Prince Henry, or Hal.


2. 
   Henry IV, part one touches on the perils of being a leader isolated from their people, and the dangers of being seduced by corrupt hangers-on [such as Falstaff].


3. I can't think of a good example for concerns here, sorry.

They had discussed touching the beauty of that place.

They discussed having physical or figurative contact with the beauty of the place. (I would interpret this as some sort of visit.)

They had discussed the touching beauty of that place.

They talked about the emotional response elicited by the beauty of the place.

They had discussions touching on the beauty of that place.

They had discussions which, among other things, involved discussing the place's beauty.

None of the examples given are correct, because "discuss" is a transitive verb, so simply takes an object with no preposition. The object is the topic of discussion.

They had discussed the beauty of that place.

touching TFD

adj

evoking or eliciting tender feelings: your sympathy is touching.

prep (OED registers use of this as a preposition as somewhat archaic.

on the subject of; relating to

Thus in AmE avoiding prepositional use:

They (had) discussed the touching beauty of (that) the place.

or

In their discussion, they had touched on the beauty of that place. (@jasonbassford)