I wonder how statement p is treated in (p AND true). Is it an open statement in this case? If it's open statement, how could we justify for the rest of the problem? (say, p AND false, why false, if p is an open statement?) I have some gut feeling about the answers in these questions but I struggle to put it in words in order to explain the logic properly.
Well, we don't a priori know that (p) is true, so we leave it depending on (p). Imagine (p) is true, then you have (true) and (true), yielding true. However, (any truth value) and (false) yields false, so (p) and (false) gives false, and (p) and (true) gives false if (p) is false.
Based on the link $P$ is defined in the problem as a statement. Thus as a statement we know $P$ is either true or false and we need not consider $P$ as an "open statement".
What $P \land T = P$ is demonstrating is something called the identity law as it relates to logical equivalences. The other identity law is $P \lor F=P$. These are called identity laws since the ultimate value is the identity of $P$... that is, the truth value of $P$ determines the truth value of the entire statement.
As for your other question about $P\land F = F$ this is the domination law of logical equivalences. Here no matter what truth value $P$ takes on the entire statement is always false so $F$ dominates.
See here for a full list of logical equivalences: https://en.m.wikipedia.org/wiki/Logical_equivalence
Given statement,
p and true = p
We don't know the value of p. It can b either true or false.
Case 1-
Let p has value false then,
From above statement,
false and true = false
As you can see the result is same as value of p.
Case 2-
Let p has value true then,
From above statement,
true and true = true
As you can see the result is same as value of p.
I guess one issue is that in that picture "$=$" is used to denote logical equivalence, as someone else pointed out. Normally we use "$\equiv$" for that, since the equals-sign is already used for equal objects. But in a sense equality also makes sense if you think of $P$ as referring to the truth value of some statement. Since we want $P \land Q$ to be true exactly when both $P,Q$ are true, $P \land true$ clearly has the same truth value as $P$, since requiring an extra true condition to hold does not affect the whole. Similarly $P \land false$ is always false, because requiring an extra false condition to hold causes the whole statement to be false.