Answers:
1.
This particular example is called a participial phrase. The whole phrase placed there is an adjective modifying the noun item. Placed is the participle, and there is the adverb modifying the participle. This example isn't actually a complete sentence, unless of course it were the answer to a question. Out of context, it doesn't have much meaning.
Example:
Which item did you touch?
The item placed there.
or
The item placed there is very fragile. Do not touch it.
2.
I'm hesitant to generalize, but I believe any past participle can be used to start a participial phrase. Note that it's not necessary to assume there is an omitted pronoun/subject, that's just the way participles work.
3.
Participle adjectives can go after the noun. Be aware that the placement may change the meaning.
I wrote to the person concerned. (I wrote to the person we're talking about.)
I wrote to the concerned person. (I wrote to the person who is worried.)
There are a handful of adjectives that can go before or after the noun, like stolen or remaining.
The stolen paintings were worth six million dollars.
The paintings stolen were worth six million dollars.
Also when describing size or age adjectives are placed after the noun.
He is six feet tall.
In general, however, adjectives in English go before the noun. Sometimes adjectives may follow the noun at the start of a more descriptive clause, but then they are usually separated with a comma.
The grass, wet with dew, tickled my toes.
The final clause defines the module you're talking about, so "that" is the proper word to introduce such a so-called restrictive clause. But there's a problem with your sentence. Since "that" refers to "module," you should be able to replace the former with the latter:
the module the exception is defined
But as your first bullet point shows, this doesn't make sense since you're missing the preposition "in."
You may say
This field specifies the module that the exception is defined or used
in.
This is slightly awkward because the preposition is stranded far from its object, the module. So you could rephrase
This field specifies the module in which the exception is defined or
used.
Or, if you're willing to personify the module as an actor on its own, taking on the characteristics of the programmer, you could say
This field specifies the module that defines or uses the exception.
Best Answer
Where there are can be with.
However, q relevant items before is a bit hard to parse, at least in this context. Does this work?