The challenge that exists in a question like this one is that the Bible is a massive text. It is a formidable challenge to know every verse and be able to hyper-analyze it for grammatical elements and logic based items. There might be some elements that I have missed which others might be able to catch.
I thought that the conjunction item could be easily found throughout the Old Testament. The Book of Genesis was one such location. A conjunction can be found in Genesis 1:6: "Let there be a dome in the midst of the waters, and let it separate the waters from the waters." A conjunction is present because a compound sentence is formed by combining two sentences with the word "and" doing so.
Continuing in the book of Genesis is an example of a conditional example of logic. Genesis 2:16 and 17 says,"And the LORD God commanded the man, "You may freely eat of every tree of the garden;[2:17] but of the tree of the knowledge of good and evil you shall not eat, for in the day that you eat of it you shall die." A conditional statement is present, even though the exact the wording is not fully reflective of a conditional statement, its intent is evident. The statement is conditional because the intent in which God is phrasing his commands reflect a conditionality in being. Essentially, God tells man that if he eats from the tree of knowledge, he will die. The "if/ then" basis of conditionality is met in the statement, as it reflects a compound statement that shows implications of action.
A statement of disjunction can be found in the book of Deuteronomy, chapter 20, verse 18. In this statement, God tells the Hebrews that they"were not to allow their enemies to live “that they teach you not to do after all their abominations, which they have done unto their gods; so would ye sin against Jehovah your God." This is an example of a statement of disjunction because God is framing a compound thought through the combination of two realities. The ancient Hebrews were instructed by God to not allow their enemies to live or they would sin against "Jehovah your God." The disjunction is where either or both sentences are true. God tells his followers to not allow "the other" to live or they will become corrupted and tainted. While there are not two separate compound sentences, the compound thought is communicated in a manner that reflects disjunction.
I found it difficult to find a pure statement of biconditionality. The premise of the biconditional statement is that something is proven to be true "if and only if" another condition is met. It is a statement of definition and emphasis. I found that the characterization of the divine in the Biblical tradition does not lend itself to creating statements that read akin to mathematical proofs. What the divine says is irrefutable. There is no additional qualification needed such as "if and only if a particular condition is met can a standard be defined. The best I could see a biconditional statement came from the Book of Ezekiel, 3:8: " But you, son of man, hear what I say to you. Do not be rebellious like that rebellious house; open your mouth and eat what I give you.” We have two statements of conditionality present. The first condition is that the Lord tells Ezekiel not to be disrespectful like the "rebellious house" to whom he is going to serve. The second condition is that his words must be the words of the divine in the idea of "open your mouth and eat what I give you." In this, one has a biconditional statement in that two conditions are reflected in the form of a definition of subservience and submission.