The simplest answer to this question is
Proofs can be comprehended at the same age rules can be comprehended.
Now, I must say why this is the case.
Although it may seem that the difficulty in answering this question is identifying what is meant by "comprehending" with regards to "proofs", the actual difficulty is in the word "proof" itself. As a human behavior "proof" is not well defined. To be more precise, there are plenty of human behaviors that can be clearly defined based on an an observable sequence of events e.g. the behavior "push a button" can be recorded and identified without the intervention of any human being. The problem with the behavior "give a proof" is that, except in the most technical circumstances, we are unable to say what constitutes the behavior of "give a proof" much less to say how we might go about observing or measuring this behavior.
(Note, it has been the goal of mathematicians since the dawn of mathematics to say clearly and exactly just what constitutes the behavior "give a proof" and so far our best efforts are found in Hilbert's formalism, but there is a huge difference between proof as argument and proof as formalism.)
Certainly, there is some agreement among humans as to what arguments are, but when we look at them closely, we are hard pressed to say why one argument "is better" or "more water-tight" than another, except to say that "we know it when we see/hear/feel/sense it."
Suppose we ignore all forms of classical Rhetoric with regards to a definition of argument. Historically, with the elimination of Rhetoric from an argument, we are left only with its "logical structure" and, it has been believed, that from this logical structure alone we can judge whether an argument is or is not "water-tight". Furthermore, we can use this judgement to say whether an argument is "good" or "bad" based on their respective use of Logic. The problem now becomes: by what standards are we to judge one logical argument as being "good" or "bad"? In other words, what rules are there to making a logical argument and how do we know when we've followed those rules and whether those rules are the "right" rules or not?
It is a common, and dangerous, misconception that mathematical statements are statements of fact; that the statement "Two plus two equals four." is like the statement "I have two pennies in my pocket." That this IS NOT THE CASE is just one of the many barriers to addressing your very important question.
The rules we use in mathematics are conventions. All that mathematics "is" is part of the statements of convention in a language. The conventions are specified by giving rules for the use of some part of the language. In arithmetic, the rules we give are the conventions for a part of a language playing the role of a number, of addition of numbers, of multiplication of numbers etc. For example, what makes a sign of a language a natural number is not its shape, color, or sound but rather the rules we follow when using that sign e.g. a sign of a language is a natural number variable if it follows the following rules:
x x
- ---
0 Sx
In fact, these rules characterize what it means to play the role of zero, the role of successor, and the role of variable all at the same time. Outside of the question "does this sign of this language follow these rules" there is no way to say whether a sign like $y$ or $z$ is on its own a "numeral variable". (As a side note, a natural number is a sign which results from the elimination of $x$ via the two rules given.)
I've said all of this so that I can conclude with the simplest answer to your question:
Proofs can be comprehended at the same age rules can be comprehended.
Now, thus far I've been limiting myself to math because there is a distinct difference between a mathematical proof, which is only and solely tested based on whether we have or have not followed a certain collection of conventional rules, and proof in the broader "legal" or "unobjectionable" or "logically correct" sense that you hinted at in your question.
The toughest thing to accept is this:
There is no experiment which can test whether a rule is "right" or "wrong".
This sounds counter intuitive at first, but the only thing we can do with a rule is to test whether or not it has been followed in a specific situation. So you might make a rule in your classroom like "Students must raise their hands before they speak." and sure enough you will observe that some students raise their hands before they speak, others speak without raising their hands, and some raise their hands and wait to be called on before they speak, but none of these observations tests whether the rule "Students must raise their hands before they speak." is "good" or "bad" or "right" or "wrong". All of these features which we ascribe to rules, specifically rules from science, are not based on any external observation, rather we choose, as individuals and a community, to say "This rule is good" or "This rule is bad".
In addressing this question we say that there is nothing which makes an argument "logical" rather we decide as individuals and as a community that "An argument which follows these rules is 'logical', i.e. 'good', and an argument that follows these rules is 'illogical' i.e. 'bad'."
So even in the more general case, when considering arguments of a less mathematical nature, the only thing required for a student to judge whether an argument is logical or not ("good" or "bad") is whether they can say whether or not a certain set of rules has or has not been followed i.e.
Arguments can be comprehended at the same age that rules can be comprehended.
The difficulty then becomes "What is the most efficient way to familiarize students with the conventional rules of our society?" and then there is the more difficult question of "Which rules should we teach as 'the conventional rules' and why?"
