Analysis Key

1. He made up his mind to win the prize, and to let the Sheriff know somehow or

other [DO of "know" that he had done so (DO)]. |

"To make up one's mind" is idiomatic for "to decide." 
     "Know" is a verbal ("The Sheriff know" will not make an acceptable sentence.) Technically, it is an infinitive with "Sheriff" as its subject. The infinitive phrase functions as the direct object of "to let."
     "To win" and "to let" are verbals (infinitives that function as the direct objects of "make up his mind." "Prize" is the direct object of "to win." [If one wishes to break down "to make up his mind," "mind" would be the direct object of "make up," and the two infinitives can be explained as adverbs to "make up" that indicate how.]

2. Nor could Robin call his men (DO) {by blowing} {on his horn}, [Adv. to 

"blowing" as he generally did, [Adv. to "did" when he was {in danger}]]. |

"Blowing" is a verbal because it functions as the object of the preposition "by." Technically, it is a gerund.

3. One day ^[NuA] this enemy came {with many soldiers} {behind him}, determined 

to kill the earl and take all his goods and lands. |

"The enemy determined to kill the earl" is an acceptable sentence, but this sentence means "who was determined." Thus "determined" is a verbal (a gerundive to "enemy"). Note also that for "determined" to be read as a finite verb it would have to be joined to "came" with an "and."
     "To kill" and "take" are verbals (infinitives that function as objects of (What?) -- or adverbs to (How?) -- "determined.") "Earl" is the direct object of "to kill," and "goods" and "lands" are direct objects of "take."

4. [Adv. to "was astonished" When the stranger saw all these fine men, dressed {in 

green}, and carrying bows and arrows, come running {to Robin}] he was very much

astonished ^(P). |

The comma that separates "men" from "dressed" indicates that "dressed" means "who were dressed." Thus it is a verbal (a gerundive to "men").
     "Men carrying bows and arrows" does not pass the sentence test, so "carrying" is also a verbal (another gerundive to "men"). "Bows" and "arrows" are direct objects of "carrying.
     "Come" will probably confuse students. "All the men come running" could be a sentence. To see that "come" is a verbal, we need to look further into the context -- "The stranger saw the men come running." Then we need the "substitution test" -- "The stranger saw them [not "they"] come running." Thus "come" is an infinitive, its subject is "men," and the infinitive phrase functions as the direct object of "saw."
     "Running" fails the sentence test since "All the men running" would not make an acceptable sentence. Thus it is a verbal (a gerundive to "men"). One could, however, also explain "come running" as a palimpsest finite verb.

5. And [Adv. to "never" because they honoured and loved the King (DO)

himself,] they would never have dreamed {of stopping him}, and {of taking 

money} away {from him}. |

"Himself" is an appositive to "King."
     "Him" is the direct object of "stopping" which is a verbal (technically a gerund) because it functions as the object of the preposition "of."
     "Money" is the direct object of "taking" which is a verbal (technically a gerund) because it likewise functions as the object of the preposition "of."