Formerly, ある could with persons or living beings as well.
(from 伊勢物語5; 全訳古語辞典 gives the meaning 昔、ある男がいたという).
It can even be used with persons today.
A paper on ある vs. いる in (modern) Japanese, where I also took the above sentence from.
There have been many copulas throughout Japan's history, such as なり(に+あり), たり(と+あり) and で+ある as well as で+ございます. Note how they are all made up of particle+ある and could be interpreted as "to exist as".
(彼は)先生である, by established usage, means "He is a teacher", a native speaker would not interpret it literally. Has a sentence with a copula such as "A is B" got a subject? The grammatical (or syntactical) subject is "A", of course, but there is no real action going on here, and so there is no semantic (meaning-wise) subject.
You can of course talk about its etymology "He exists as a teacher.", and here he is doing the existing, as a teacher.
で居る, on the other hand, is not a standard copula, and is interpreted literally.
(lit. "You will/can feel happy when you exist with a smiling face", ie "Smile and you will feel happy.")
Before I get to 先生でいる, the 広辞苑 gives this helpful insight on 居る:
It once meant 座る, とまる or 住む as well.
先生である focuses as something (or someone) being (defined as) a teacher. On the other hand,
focuses on acting as or like a teacher.
As for 感じがする, it is again important to distinguish between syntactical and semantic subject. It might prove to be of interest to inquire the etymology of this idiom, what the people were originally thinking (about) when they said this. I can imagine that, as a metaphor, they really ascribed an active role to their feelings, similar to English "my emotions are over-whelming me."