3. How did the preaching about the Kingdom and the healing of diseases work together in their ministry? Luke 9:1-10?

The "preaching" was not about the "kingdom", but rather about salvation and their message was supported with miracles, signs, and wonders to prove their message was true (something you don't see any of the cults doing).

For me I see that Jesus Christ was sent to establish God's Kingdom in The Way which pleased God most. For instance, the religious (Pharisees taught > that if someone was ill or diseased this was due to some sin committed; the Jews believed that if someone was ill or diseased they had sinned in some manner; and this belief certainly was proved wrong by Jesus Christ who pitied the multitude, who were like harassed and panting sheep without a shepherd, and like a harvest left unreaped for want of labourers (Matthew 9:36-38). 

THE LORD GAVE THEM SPECIFIC INSTRUCTIONS SO THEY KNEW WHAT TO DO. HE TOLD THEM HOW TO DEAL WITH TOUGH TIMES (9:5) AND HELD THEM ACCOUNTABLE.

Read the scripture for yourself to get the answer - this is your homework, not ours.

Why is this question numbered? Are you asking for homework help?

Joh 20:30  Now Jesus did many other signs in the presence of the disciples, which are not written in this book; 
Joh 20:31  but these are written so that you may believe that Jesus is the Christ, the Son of God, and that by believing you may have life in his name. 

Mar 16:20  And they went out and preached everywhere, while the Lord worked with them and confirmed the message by accompanying signs.

Act 2:22  “Men of Israel, hear these words: Jesus of Nazareth, a man attested to you by God with mighty works and wonders and signs that God did through him in your midst, as you yourselves know— 

Heb 2:3  how shall we escape if we neglect such a great salvation? It was declared at first by the Lord, and it was attested to us by those who heard, 
Heb 2:4  while God also bore witness by signs and wonders and various miracles and by gifts of the Holy Spirit distributed according to his will. 

The miracles proved to their listeners that the speakers spoke for God and thus could be trusted (ref Ex 4:1-9).  Their messages were documented resulting in what we now call "The Bible".

For curiosity's sake, update your question with how many questions did your teacher give you, what class you are taking, and when is your homework due.

16 "He that believeth and is [Holy Spirit] baptized shall be saved; but he that believeth not shall be damned.

17 And these signs shall follow them that believe [That one doesn't receive after a water baptism; Matthew 28:19;] In my name shall they cast out devils; they shall speak with new tongues;

18 They shall take up serpents; and if they drink any deadly thing, it shall not hurt them; they shall lay hands on the sick, and they shall recover.

19 So then after the Lord had spoken unto them, he was received up into heaven, and sat on the right hand of God.

20 And they went forth, and preached every where, the Lord working with them, and confirming the word with signs following. Amen" (Mark 16:16-20)!

“And he answered and said unto them, I tell you that, if these [Today's Christian] should hold their peace, the stones would immediately cry out.” (Luke 19:40)!

The latter simply confirmed the former, Heb. 2:3-4.