I could tell it was "her" right away. However, I have no idea why it is used "vient" (present tense) when the English translation is "lost" (past tense).
In French, "venir de + infinitive" translates "to have just done ....(the action the infinitive verb indicates)." So,
"Je viens d'expliquer" translates as "I just explained" "Vous venez d'arriver" as "You just arrived"
I think 'il vient de faire quelque chose' means 'he just made something'? Can anybody explain?
Yes, that's right. the English construction with "just + preterit verb" does not exactly translate in French, so you have to use the typical French construction "present venir de + infinitive verb".
I put "a friend just lost their job", sure, the gender is clear in french, but my sentence doesn't rule out that the friend is not female. Should this be marked as right?
On Duolingo "their" is conventionally translated to/from "leur/leurs", meaning "2 or more owners".
You need, therefore, to translate this sentence with "her job", since "une amie" is necessarily feminine.
And we do have subject/pronoun agreement in English, even though your translation is often used. Since the subject is singular feminine, the possessive pronoun should agree. Strictly speaking, your translation means: My friend just lost the singular job of two or more people.
Is there a specific reason why "A friend came to lose her job" is not accepted? "came to be" or "came to know" are valid constructions in English but I wonder if the tense is much different from what seems like the literal English translation.
"Came to be" or "came to know" are valid constructions in English, but not the translation for the French sentence.
A back translation for "a friend came to lose her job" can be "une amie en est venue à perdre son emploi".
You are familiar with the near future (futur proche) which says "Je vais écrire une lettre" (I am going to write a letter). This sentence demonstrates the near past: "Une amie vient de perdre son emploi" = A friend just lost her job. When a conjugated form of "venir de" is followed by an infinitive, it means that the action of the infinitive just happened recently.