Yes, I don't know of any prohibition about Mormons being Masons. 
 
Yes, divorced persons can become Mormons. Also - just to clarify - we do allow divorces within our church. It's obviously discouraged and much effort is put in place to help struggling couples, but divorces are allowed.

Q) Can Mormons belong to the masonic order? 
 
A) Absolutely, Joseph Smith Jr. the founder of the Church was a Mason. After the Prophet's death there was a ban placed by the Masons on accepting Mormons into there order. That ban though ended many many decades ago and since then Church members have been members of the Masons.  
 
Q) Can a divorced person become a Mormon? 
 
A) Yes, divorce does not bar someone from entering the Church rolls nor does getting a divorce effect one's standing in the Church unless the was some other transgression committed (like adultery or spousal abuse) that caused the divorce.  
 
One important thing to remember though is that while a marriage "for time only" by a civil authority can be ended, the temple sealing or temple marriage cannot be undone in civil courts.  
 
A member who civil divorces there spouse is still sealed to there spouse fro time and all eternity if they were married in the Temple, and they are prohibited from remarrying in the temple. The only way around this is to petition the President of the Church to have the sealing removed. At the point the member can remarry in the temple. 
 
Also it is important to note that if you have children with a previous spouse you will not stay a member in good standing if you owe any back child support, temple recommends, priesthood offices, and callings are all dependent on the member fulling there legal requirement to pay child support for there children.

If one of the spouses decides to leave the church and the result ends in divorce, what happens to the children of the couple? Are the children considered mormon? Are the children still accepted in the church? Is it accepted in the church for the continuing mormon to reject those children?

Yes, as I understand the children are still members of the Church. Once a person is baptized it requires a written request from the individual to have their names removed. Given the parents situation - I believe it would require the written permission from both parents to have the childrens names removed. 
 
Yes, the children absolutely would be welcome. There is no such thing as shunning in the church.