Masato, these are some amazing examples! Thanks, very much. I wondered if it didn't have something to do with marbles, and I looked at several websites, but only found references to the "taw" as the "shooter", and to the "taw line" as the position from which one shot his marble. To "bring to taw" or "come to taw" as a return to the starting point would certainly make sense as a name for a race horse! Perhaps we would say, "it will all come back around at some point," or "what goes around, comes around". Or simply, "to come around" to seeing something, or agreeing with something. Your second example is especially interesting in relationship to the verse from "Limber Jim" with regard to the second line which is "we won't have any more of your link and law" or perhaps "Lincoln Law". In this case, the reference to "snakes" may be more like "snakes in the grass" or some other derogatory term, and might have had a specific historical reference at some point. The verse is interesting because it is so different from the other three.
I would be interested to know if "taw" in this sense comes from England or Ireland, and is it still in current use in relation to marble games? And if anyone has heard the phrase used otherwise - "come to taw".