Find the subject of the verb "existed"

Pregnancy has stirred up feelings I didn't know existed!

I guess the subject of existed is feelings and I kind of understand the meaning but I can't work out the structure of this sentence.

• Where is the subject of existed?
• Why is it "ed"?
• Is there any grammatical rule for using this structure?

Yeah, you've hit on a weird quirk of English grammar here (modern German does similar things, I think).

You have three ideas here.

1) Pregnancy has stirred up feelings.
2) The feelings existed.
3)I didn't know that fact.

Combining the last two, you get

"I didn't know that the feelings existed."

You can leave out the subordinating conjunction "that" in English and get "I didn't know the feelings existed."

Then you can turn this sentence into a relative clause with the first sentence: "Pregnancy stirred up feelings that I didn't know existed" where the "that" is the relative pronoun replacing "the feelings" in "the feelings existed," and "that" is the subject of "existed."

Finally, in English, you can drop the relative pronoun "that," And you wind up with your sentence. Oddly, the real subject of the "existed" is the second "that" that dropped out. "Feelings" is the direct object of "stirred up," and the actual subject of "existed" has disappeared in the flesh, but is there in spirit.

The -ed is simply the past tense.

This is a weird feature of English (and general Germanic grammar, I guess).

It arises when you have a statement like these, where you have a noun clause beginning with "that" as a direct object of some verb of thinking, saying, perceiving, or showing:

 I know that the girl eats fruit.
 I think that the boy likes grapes.
 He says that the fish is blue.
 We showed that the bird was singing.

These can be rewritten without the subordinating conjunction "that":

 I know the girl eats fruit.
 I think the boy likes grapes.
 He says the fish is blue.
 We showed the bird was singing.

You can now make a relative clause with the relative pronoun "that" replacing the subject of the "that" clause.

We see the girl that I know eats fruit. 
I met the boy that I think  likes grapes.
Sarah caught the fish that he says is blue.
She recognized the bird that we showed was singing.

As often, English can then omit the relative pronoun "that":

We see the girl I know eats fruit. 
I met the boy I think  likes grapes.
Sarah caught the fish he says is blue.
She recognized the bird we showed was singing.

In all these cases, the disappearing relative pronoun "that" is the real subject of the verb towards the end of the sentence (eats, likes, is, was singing).