I have trouble with article usage when I mention some activity I do in a specific place which has a proper name.
This play was presented at the La Scala Opera House.
Do we need the definite article there?
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In your example sentence, the actual name of the building is (in English anyway) simply La Scala.
If you use only a proper noun, you do not normally use an article. (Although, stylistically, you might want to in some cases.)
In your example sentence you could say:
The play was presented at La Scala.
You don't need the article because you're referring to the actual name of the building.
If you use just a common noun, you do need to use an article:
The play was presented at the opera house.
The play was presented at an opera house.
But if you want to use both a proper noun and a common noun, then you use the definite article—and only the proper name is in uppercase:
The play was presented at the La Scala opera house.
But let's imagine that the actual name of the building were, literally, La Scala Opera House.
If this were the case, then Opera House (what would normally be a common noun) would actually be part of a proper noun.
Do you put an article in front of it or not?
There is no completely correct rule for this.
In theory, you could say:
The play was presented at La Scala Opera House.
Or you could say:
The play was presented at the La Scala Opera House opera house.
But neither of those necessarily seems correct because of how we're used to constructing similar sentences where this kind of confusion doesn't exist.
I suspect that most people would use an article with only the proper name—because it would seem correct even though, technically, it doesn't fit one of the normal uses:
The play was presented at the La Scala Opera House.
Note, however, that this is not actually the case in your example sentence. The name of the building is simply La Scala. So, refer to the first part of this answer.
The following link may be of some help in learning about the use of articles before nouns, Articles, Determiners & Quantifiers
To answer your specific question - Properly called Teatro alla Scala, the theatre is most commonly called by the much shorter name La Scala. In English La Scala means The Staircase, so there really is no need for the definite article because there is already a definite article in the theatre's name.
So, if you are talking about La Scala, you usually leave the definite article off. For stylistic reasons, if you think that a particular sentence will sound better with the definite article (i.e., 'the La Scala'), it is not wrong to include it. However, if you are using the full name of the theatre (i.e., the Teatro alla Scala), then you would usually put the definite article in front of it.