To understand the relationship between fallacies and validity, you need to first understand the fact that an argument (a pair set of premises and a conclusion) has form and content. The form is concerned with the logical schema between the conclusion and premises; the content is concerned with the reasonable support (or persuasion) of the conclusion by the premises. The two are generally studied independently: the form, in logic; the content, in rhetoric.
A fallacy is a defect in the argument. The defect can happen in form or in content. When the form is defective, the argument is called as committing a formal fallacy. Affirming the Consequent or Denying the Antecedent are examples of the formal fallacy. Committing formal fallacies always produce invalid arguments. When the content is defective, the argument is called as committing an informal fallacy. Committing informal fallacies, however, can be logically valid as in circular reasoning (petitio principii) since the conclusion is already contained in the premises.
Based on this understanding, let's examine the three questions (or statements) of yours.
- If an argument commits a fallacy, it is invalid.
No, If an argument commits an informal fallacy, it can be valid. e.g., petitio principii
- If an argument is invalid, does it commit a fallacy?
If an argument is invalid, then it must commit a formal fallacy.
- If an argument commits a fallacy, must the conclusion be false?
The conclusion of an argument that commits formal and/or informal fallacies can be still true.
Denying the Antecedent (formal fallacy)
If I am God, I can eliminate all sufferings.
I am not God.
Therefore I cannot eliminate all sufferings. (True concision)