I can’t fill you in on this because I can’t remember, the addresses are not kept on file.

Евгений, FILL IN is different. That’s when someone comes into a situation after it’s already started and you bring them up to date.

*I can’t TELL you because I CAN’T remember; [no article] addresses aren’t kept on file. 👍🏼

I can’t tell you because I don't remember. Addresses weren't saved in a story.

Uly, why [no article] before "addresses". It's asking to be there...

No it isn’t. THE addresses would be a specific set of addresses, ADDRESSES alone refers to ANY AND ALL ADDRESSES

I know that. But from the original it seems like the speakers both know what set of addresses they are talking about. Why do you think they are talking about the some random addresses?

Because they’re saying that IN GENERAL, addresses aren’t stored in the history.

Compare: “Our uber drive picked up five people today, but unfortunately THE addresses weren’t kept on file.” Here they’re talking specifically about the addresses of those five people.

Compare: “On the uber app, addresses are stored in history so you never have to retype them when you go back to a place you’ve been to before.”

Can you tell me all addresses you have visited within the last 2 weeks?
I can’t tell you because I can’t remember - THE addresses aren’t kept in GPS history.

to me the original sounds like an answer to a question where the addresses were mentioned, like I wrote above.

*Can you tell me all THE addresses you’ve visited within the last 2 weeks?
I can’t tell you because I can’t remember - ADDRESSES (=as a rule) aren’t kept in THE GPS history.

Little confusing. To me (and to Eugene, based on his translation) "the" was asking to be there.
I wonder if there are some solid rules (tips) that would allow foreigners to use THE/A always correctly. I know there is a set of rules but often in real life, I find situations when going thru all rules doesn't help me and I would sitting and thinking between THE or no THE . Any magic tips?

Nope, this question is too nuanced. But in this case, I can repeat - the lack of an article with plural refers to any and all instances of that noun: PATIENTS must wear a mask at all times. TEXTBOOKS are to be paid for in advance. ADDRESSES aren’t stored in the history, so be sure to write them down if you need to remember them.

Uly, have you ever been confused with using articles? Has it ever happened to you that you would be thinking for a second whether to use THE, A or none?

Never ever, thank God. English and Spanish both use articles, so they’re second nature to me.

do we always call them Articles? is there another word for them?

They are definite and indefinite articles.