What kind of adjective is the word 'ALONE'

Yes, you can indeed use "alone" in this way and with no comma required. Here is a rather nice quote from Dragon Lord Messiah (Elijah Stone): "Yes. That is how I must remain. To truly be true to myself, I alone must stand, alone." Two uses of the word in one sentence (one predicative; one not)! :)

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 myself

by yourself

by itself / himself / herself

by ourselves

by yourselves

by themselves

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).

BillJ posted the following answer in the comments:

Your examples are all fine: you are right that "alone" is a post-head modifier in noun phrases, but note that it is an adverb here, more precisely a focusing adverb.

As a post-head modifier it is an adverb, not an adjective. When it is an adjective, it functions as a predicative complement in examples like They felt very alone on the moors that night ; Kim prefers to be alone; They were alone when the burglars struck.

Huddleston & Pullum's The Cambridge Grammar Of The English Language gives "alone" as a focussing adverb on p587 and discusses it in depth on pp591-2