Yes, the plates are no longer on the Earth the Angel Moroni who buried the plates in what is today Palmyra, New York in 421. Revealed to the Prophet Joseph Smith the location of the Plates in 1823 he had him return on the same date for the next 4 years to receive further instructions and prepare to receive the plates.  
 
The Plates were received by the Prophet in 1827 and translated in the spring of 1829. The reason why it took so long for the translation to start (almost 2 years) is no soon as the plates were taken from the Earth wicked men were conspiring to steal the plates and destroy them. The Prophet was consistently moving and hiding just to try and keep the plates secure. He even had to flee New York into Pennsylvania in order to get the translation started. From there it only took about 3 months to complete the translation.  
 
This is the reason why after the translation was complete the Angel Moroni took the plates back and removed them from the face of the Earth, wicked men were still seeking to destroy the plates and prevent the restoration of the Gospel of Jesus Christ, just like today wicked people still try and thwart the growth of the Church and the message of the Restored Gospel around the world.  
 
Had the plates remained they would of been destroyed and the plan of the Lord frustrated. It was a miracle that they were not destroyed in the short span of time they spent in the hands of the Prophet given how violently they were sought to be destroyed.  
 
Heavenly Father does want the world to know the truth of the Restored Gospel of Jesus Christ but this knowledge cannot come from the plates that are in Moroni's safekeeping. Believe in the Gospel does not come from sight but from the Holy Ghost testifying to one's soul that the message of the Book of Mormon is the Word of God, written by ancient prophets in the Americas.  
 
To answer your third question yes three was 12 witnesses to the existence of the Plates. 4 of the Witnesses actually saw the Angel Moroni and he spoke with them and told them of the truth of the origin of the plates. Another 8 witnesses were allowed to view the plates and they testify that the plates do exists.  
 
These Witnesses testimonies are found in the introduction of the Book of Mormon which is available to be read online for free at http://lds.org/scriptures/bofm/introduction?lang=eng  
 
You can ask many people about this and receive many answers as well with any questions about the Church, including from myself. The best source for official information about the Church of Jesus Christ of Latter-day Saints is our websites lds.org and mormon.org 
 
I hope that my answers helped you understand the subject more clearly and brings the love of Christ into your life more abundantly!

Just an interesting thing to note about the 3 & 8 Witnesses that LDS Guy references. 
 
All of the 3 witnesses - and some of the 8 witnesses - became disaffected from the Church. Some even left the Church. You would think that if they left the church that they would disavow their testimonies and say that what they had seen was not true. But every single one of them - to their very last breath - stayed true to their witness and testified to the very end that what they had seen - and in some cases felt - was true. 
 
Again - here is the link to their witnesses and testimonies: http://lds.org/scriptures/bofm/introduction?lang=eng