Why is “If” sometimes followed by “were” as in “If I were the President, I would cancel all exams.” [Fat chance, but no harm in dreaming, is there?]
That’s it! In the conditional clause “If” is followed by “were” when the idea is improbable.

Read these sets of sentences to see the difference in normal and improbable situations.
[1] Normal: If I knew her name, I would tell you.
Improbable: If I were you, I would report this at once. [I cannot be you. I just assume something improbable.] “Were” is used to show the possibility of something that could have happened. It has the sense of “Ah, if only it had happened that way!”

[2] N: If I am elected president, I will reduce taxes. (Said by a presidential candidate)
Im: If I were elected president, I would change the school system. (Said by a schoolboy)

N: If we win this match, we will qualify for the semifinals.
Im: If I won a lottery, I would stop teaching. (improbable)

[3] N: If dad had divided the property (then), I would not be a pauper (now). Here we are talking about a situation that did not happen in the past.

Summing up
The conditional construction does not normally use will or would in if-clauses.
Exception: [1] Request: If you will come this way, the manager will see you now.

[2] Willingness: I would be grateful if you would give me a little help.

For improbable situations, use “were” with “If”. If I were a rich man…

Finally, the mixed sentence: If she had written to me [then], I would not be married. [now].

Why do we use “were” for a singular subject (If I were…)? That is because were is not a plural here. We use “were” to show the subjunctive mood – to show this is unlikely, hypothetical. It is something we imagine. “Were” is the indication of that.
If the situation was true, you simply use “was”. If I was doing well, it was because of my aunt.

Error here:
Incorrect: If the Senate confirms her after hearings that begin on June 28, she would be the 112th justice of the United States Supreme Court.
Correct: No conditional form is needed here: “If the Senate confirms her, she WILL BE the 112th justice.”