I really don't know the scope of the grammatical rule cited in the OP, but judging by the usage in the King James Bible (published in 1611, closer to the time that thou/thee/thy/thine was an active element in English), although thine own seems to have been the more commonly used phrase, thy own was not unknown. For example, Luke 2:35:  Yea, a sword shall pierce through thy own soul also ...

According to Barber's Early Modern English the rule mentioned in the OP (adding that thine was used before some words beginning with h) applied "pretty consistently" during the earlier part of the 16th century, but by 1600 my/thy and mine/thine are "in free variation before vowels". He gives the example of Shakespeare's use of both "thine eye" and "thy eye". (There is more recent research taking issue with the "free variation" description, and proposing other interpretations, but I don't think there is consensus on how the usage should be described). Barber goes on to say, however, that by 1700 the "thy" form was almost always used in standard literary prose, "thine" was an archaism. So the lyric aligns with the usage of the day.

And the opening line of The Battle Hymn of the Republic, viz "Mine eyes have seen the glory of the coming of the Lord", written in 1861, would have certainly have been an anachronism.

Re #4. Yup. All is fair in poems and pastiche.

The author of the blog below, John Wells, thinks J. W. Howe might have been echoing the words of Psalm 121 in the Prayer Book, I will lift up mine eyes unto the hills.

https://www.phon.ucl.ac.uk/home/wells/blog070...