"I am happy that I met you."

And also in "I am happy with your success."

With your success modifies I am happy. Do you agree with me? Thank you so much.

AnonymousI think that "that I met you" modifies "I am happy".

No, I don't think so. It is just a dependent clause, with the conjunction "that." Here is a similar structure:
I am happy because I met you.

Hi Alphecca Stars
Please help me with this:
'I am happy that I met you'
Yes, with because the sentence appears easy to analyze. Is the sentence grammatical with that as conjunction?
If it is grammatical, is the structure (with that) idiomatic?

If it is grammatical, is the structure (with that) idiomatic?

It is grammatical. That can be a conjunction. See entry #13
http://dictionary.reference.com/browse/that?s=t&ld=1120

I went through it. Thank you Alphecca Stars.

Please help me with this:
She said that he would come.
(Here that is the conjunction and that he would come is the subordinate clause.)
She was happy that he would come.
( Here, is the subordinate clause object of the complement 'happy'?)

According to this reference, yes. Note that it says "some analysts" consider this construction to be a complement clause.
Others might just call it an adjunct.

http://www.sil.org/linguistics/GlossaryOfLinguisticTerms/WhatIsAComplementClause.htm

Yes. I am happy that I met you, too.