Do these constructions have a standardized way of being translated into English, or can their translation vary based on context?

Depending on context, you may prefer to translate “there is (are some)/there isn’t” with “il existe/il n’existe (aucun)”, especialy in maths.

There are some idiomatic sentences using it that may more or less idiomatic, like “il n’y a qu’à voir” that you may prefer to translate “having a look at it say it all” or something like that.

Also, in a more familiar register, “(il) Y en a, je te jure…”, may be translated “Some people make me feel so angry, I swear that would they be in front of me…”.

But most of idiomatic sentences using this construction are idiomatic on an other part of the sentence, like “Il n’y a pas de quoi en faire un fromage !”, that is “There’s nothing to make a drama out of”. Well you may just as well say “Il n’y a pas de quoi en faire un drame !”, but you know that frenchs and cheese is an epic story. 😉

We were taught that “il y a” means “there is” and “il y en a” means “there are”. However, that was in the early eighties – and I did have an excellent French teacher, but maybe modern French is changing dramatically enough to drop subtle ‘formalities’. Again, it may depend on the context in which it used. I hope this is of some help.