Alone is an adverb, not an adjective. (The adjectival form is lone, as in The Lone Ranger.) "Alone" is equivalent in meaning to the following adverbial phrases, but being one word only, "alone" is preferable to
by itself / himself / herself
Note that adverbs and short adverbial phrases do not need to be set off by commas.
Substitute the above mentioned adverbial phrases for the word "alone" in the three example sentences you supplied, and you will see how they modify the verb--not the noun. Hopefully you also see why a comma is unnecessary, and see that the single word "alone" flows more smoothly than the phrase.
For the predicative case, consider the two equivalent constructions,
I went home alone.
I went home by myself.
But the meaning would be different if I said,
I alone went home.
The meaning in this case is that I was the only one who went home (while others stayed out / at the scene / away from home).