In the US we typically say that classes are offered rather than given
The preposition will be by or at, not from.
What is offered is more likely to be courses than classes. Classes tends to be used more in secondary-school or non-academic contexts.
So:
English courses for people whose native language is not English are offered by that university.
But note: the passive is awkward in this instance. Other things being equal, English prefers to put “heavy” constituents—those with many words—at the end of a sentence rather than the beginning or middle, because the syntactic structure is most visible when the primary constituents are close together at the front. Compare the structures for the passive and active sentences:
[Subj English courses MinAdj for people whose native language is not English] [Verb are offered] [MajAdj by that university.]
[Subj That university] [Verb offers] [DirObj English courses MinAdj for people whose native language is not English.]
In the passive sentence, that's a long run from Subject to Verb. In this case you could get around this by moving the “minor” adjunct to the end:
English courses are offered by that university for people whose native language is not English.
[Subj English courses] [Verb are offered] [MajAdj by that university] [MinAdj for people whose native language is not English.]
Number (2) is fine, correct and natural.
Number (3) should say, 'Some money was given to her.'
Number (5) is close, but the 'to' should be replaced with 'by him': 'She was given some money by him.' However, you could omit both so it's exactly the same as Number (2).
Number (6) is fine, correct and natural.
As for the best way to get this into the passive voice, I would say 'She was given some money by him' is your best option.
Best Answer
Both of those are completely acceptable and correct.
Edited to add: What happens is, there are two separate correct ways to phrase this in the active voice:
This alternation is called "dative alternation" or "dative shift".
I should mention, by the way, that in this particular case, "He gave a test to me" is rather awkward; generally we only use that variant when the object of to is "heavier" than just me: "He gave a test to Jill, Bob, and Susan."
Since English frequently allows any type of object — direct object, indirect object, object of preposition — to be "promoted" to subjects via passive-voice transformation, we have four options that might seem possible:
However, *"I was given a test to" runs afoul of a restriction on prepositional passives: they don't work when there is also a regular direct object. (For example, we can say "this bed has been slept in", meaning "[someone] has slept in this bed", but not *"this bed has been put kids to sleep in", meaning "[someone] has put kids to sleep in this bed".)
The other three possibilities are all correct, though the "[…] was given me" version is sufficiently uncommon nowadays that I would not recommend it for current use, at least in U.S. English. (It will be understood, but it will sound odd, at least to American ears. British ears may feel differently.)