We need some conventions on the terminology.
A sentence
is a meaningful group of words that express a statement, question, exclamation, request, command or suggestion.
A declarative sentence (stament, assertion) is a sentence stating a fact, like: "The rose is red".
In logic, there are declarative sentences and not e.g. questions or commands.
Unfortunately, we call the propositional calculus also sentential logic.
Thus, in propositional calculus we can replace a sentential variable $p_i$ with the declarative sentence: "The rose is red" and not with the question: "Which is the color of the rose?"
In propositional calculus, sentence and proposition are interchangeable, while in philosophical discourse, a proposition is usually an extra-linguistic entity: the content expressed by, the meaning of, the reference of a linguistc entity (a declarative sentence).
In predicate logic we have formulas with free variables (called open formulas), like: "$x$ is red".
The free variable acts as a pronoun; when we assert "it is red", the meaning of the assertion is disambiguated contextually: according to the context, "it" may stand for the book on the table, the car, the pen, and the truth value of the assertion may change accordingly.
In logic, we have to assign a denotation to the free variable $x$ (through e.g. a variable assigment fucntion) in order that the formula has a meaning (and a truth value).
In predicate logic, a formula without free variables (or closed formula), like "the rose in my hand is red" or "all the roses are red", is called a sentence.
Thus, in conclusion, either in propositional logic or in predicate one, a sentence is always meaningful, and it has always a definite truth value.
The first thing I want you to keep in mind is that there are really two somewhat distincts uses of a truth-table.
First, truth-tables are a way to display the semantics of truth-functional operators. For example, we define the material conditional $P \to Q$ to be a certain truth-function: a function that tables in truth-values, and that outputs a truth-value. We could describe that function as follow:
$\to$ is a function that takes in 2 arguments, and each argument is a truth-value: something that is either True or False. $\to$ will output one if those two truth-values as well. Specifically, $P \to Q$ is False if $P$ is True and $Q$ is False, and it is True otherwise.
Now, that is a perfectly good mathematical definition. But we typically display ('specify' if you want) the function by this truth-table:
\begin{array}{cc|c}
P&Q&P \to Q\\
\hline
T&T&T\\
T&F&F\\
F&T&T\\
F&F&T\\
\end{array}
So this is just a little more organized way of showing how the $\to$ works as a truth-function.
Now, you ask: do we prove that the table for the $\to$ looks like this? No, we don't. We simply stipulate it. We define the $\to$ to work this way. Of course, the intent is for $\to$ to try and capture certain aspects of English 'if ... then ...' statements, and we can debate how good of a good it does in doing so. However, as far as pure logic is concerned: it's just something we define ... and we'll leave it up to you to see if there is some kind of useful application for it. But no, we don't prove our definitions.
So, is it an axiom, you then ask? Well, I don;t know if 'axiom' is quite the right word here, as within the context of logic we typically view axioms as statements that express what is considered some elementary logical or mathematical truth. Again, it is more of a definition, or maybe an assumption: it is part of laying out a system of truth-functional logic.
A second use of truth-table is this: once we have defined our basic operators like $\neg$, $\to$, $\land$, and $\lor$, we can also use truth-tables to investigate the truth-functional properties and relationships of more complex statements. We can, for example, 'work out' the truth-conditions of the statement $P \land (Q \lor R)$, and compare those to the truth-conditions of $(P \land Q) \lor (P \land R)$. And, lo and behold, we find that these two expressions have the exact same truth-conditions, and thus we say that the two statements are logically equivalent.
Again, though, we don't prove this table either. You could say that the table proves the distributive property, and the table itself is not proven. Rather, the table shows what happens to the truth-conditions of these two sentences once we accept the operators that are involved to work a certain way, and as explained in the first half of the post, we already assumed that they work a certain way, like it or not.
Best Answer
In these examples, what they mean is that the variable $x$ does not point to a fixed object (in this example, the context has not supplied $x$ with a fixed value), so formulae like '$x$ is greater than $5$' are merely propositional functions rather than propositions/statements.
(Recall that a function has a varying output that depends on its input. Do read the linked answer before continuing below.)
In these examples, they are writing 'if $x$ is greater than 5, then $x-5=0$' (which technically isn't a statement) as a shorthand for 'for each $x$, if $x$ is greater than 5, then $x-5=0$' (which is a statement). This is called implicit universal quantification.
Does "Ram" refer to the particular person that you have described, who is distinct from other persons also called Ram? If so, then as you have pointed out, the above is a true statement.
On the other hand, if "Ram" is understood to be an arbitrary member of the collection of all persons named Ram, then we are reading 'If Ram is a man then Ram has beard' (which technically is not a statement) with an implicit universal quantification, that is, reading it as 'Every person named Ram who is a man has a beard'; in this case, the statement indeed needn't be true (and in our universe is likely false).