You're correct that it should be the current year if your speaking about this year. The only thing I can think of is that they were using 'current year' as a place holder. "How did The Sims 2 predict rent charges in [current year]? " i.e., 'in that year' or in 'a given year'. Did they talk about rent prices in different years? If you post the link I can have a look.

It's a bit slang. It's a working class dialect. Some people speak like that always. It's fine in casual speech, because everyone understands it, and they appreciate it's part of his accent and dialect. The in a formal situation, like if you're speaking to royalty, or are in an exam, don't omit that word.

Sims is a game which simulates moving through time. ‘Current year’ means ‘whatever year it is in the game’ or, more formally, ‘the game variable indicating the current year’. In writing, of course, the YouTuber would have to do something to indicate that it IS a variable name, using things like capitalization, running the words together, different fonts, or quotes (‘CurrentYear’ e.g.) In speech, though, that information is conveyed simply by omitting the ‘the’.

Yes, you're right. It's not standard English to omit the article in this case. It may be some sort of shorthand or a regional or individual variation, but it would be graded as wrong in any kind of formal testing. The rules of grammar tell us that we always need an article or some other determiner before a singular countable noun; and in this case, it needs to be the definite article "the" because he's talking about a specific year.

You are right that it would be better to use an article. However, I would not consider it to be an actual mistake to omit it. Articles can be omitted in order to use a noun as a category rather than a specific thing: "Make yourself feel at home" "We did all that work for nothing" "My favorite food is chocolate" "I don't like to eat salt" What makes "current year" sound somewhat wrong is the fact that it is specifically a countable noun. It is harder to use countable nouns without articles, but it is still possible: "He won first place." "She arrived at the party without boyfriend" This last example is often written in English with the imported French word "sans" which never uses an article: "She arrived at the party sans boyfriend"