As one comment says, your first example is fine, but it's not really an example of what you're asking about.
If you're wondering if this:
In order to keep the colony size down, a predator, which keeps the colony size down by eating bacteria, is introduced.
means the same as this:
In order to keep the colony size down, a predator is introduced, which keeps the colony size down by eating bacteria.
Not exactly. They both work because both "a predator" and "the introduction of a predator" can be said to keep the colony size down.
If "a predator" were followed by something unrelated to keeping the colony size down, then it wouldn't make sense semantically. For example:
In order to keep the colony size down, a predator is drinking from the river, which keeps the colony size down by eating bacteria.
Your second example isn't fine, because unless she put forward that the idea was interesting, it is indeed out of order. If I understood your question, you are asking if:
The idea, which she put forward, was interesting.
and
The idea was interesting, which she put forward.
are the same in meaning. And they are not.