1. I agree with you that watching TV is a waste of time.
2. I agree with you about the statement that watching TV is a waste of time.
3. I agree that watching TV is a waste of time.
-----------------------------
Q1: Are they all grammatical? Do they have the same meaning?
Q2: What is the part of speech of 'that' in Sentence 3? Is "the that clause" an adverbial clause in #3?
All 3 mean about the same thing, although #2 is wordy.
The word "that" in #s 1 and 3 is a subordinating conjunction except it's introducing a phrase instead of a clause.
http://grammar.ccc.commnet.edu/grammar/conjunctions.htm#subordinating_conjunctions
A1: Yes, all three sentences are grammatically correct and have the same meaning. They express agreement with the statement that watching TV is a waste of time.
A2: In Sentence 3, "that" is a conjunction which introduces a subordinate clause, also known as a "that clause." The "that clause" functions as the object of the verb "agree." It does not function as an adverbial clause in this sentence. Instead, it clarifies what the speaker agrees with, which is the proposition that watching TV is a waste of time.