We say neither:
for two reasons. These "sentences" sound far more strange and incorrect to native speakers than you could probably imagine.
While the "grammatical" passive-voice form of 「やる」 is certainly 「やられる」, the latter generally has a fairly negative connotation. "To have something undesirable done (...
You should say この花には水がやってある or この花は水がやってある.
But この花は水をやられた and この花は水がやられた may be grammatically correct.
この花は水がやられた may sound like another flower may be poured a kind of liquid apart from water to.
Both mean “My arm was caught.” But second one sounds like Another one’s leg or hand may have been caught.
I don't know what それ refers to without reading the previous context.
The basic structure of this sentence is AをBと思い込む, which means "to (wrongly) assume A as B", "to make a false assumption that A is B", "to convince oneself that A is B", etc. See this answer for the A + を + B + (だ)と + verb ...
No, there is no rule that you always have to use the passive form.
However, I realized that a Japanese sentence often omits the subject, often implying the subject is the speaker without other context, and that this style makes the sentence in the passive form.
He told me what I needed to know.
can be translated to
I assume the question used the versatile word "let" to mean the teacher becomes the decision maker by choice, and not by command, which is otherwise rude.
Personally, I would avoid the direct translation. The following suggested expressions use 私のグループは that refers to the group I would be in (translated as "my grouping"); not the literal "my group", which is ...
In the specific example you give, I'd probably just say 「私のグループは先生に任せた」 ("I left my group up to the teacher") or similarly 「グループの選択は先生に任せた」("I left the choice of my group to the teacher").
In the case you provide, it's more like you're saying you're leaving something up to someone else, rather than allowing someone to do something to you.
If you're ...