Why, if all are separated from God because of Adam's sin, did God choose Noah? Why be merciful to Noah's descendants if all are separated by sin?
Genesis 6:5-6 describes God’s view of the people who lived prior to the flood:
“The LORD saw how great the wickedness of the human race had become on the earth, and that every inclination of the thoughts of the human heart was only evil all the time. The LORD regretted that he had made human beings on the earth, and his heart was deeply troubled.”
God determined to wipe out the evil deeds of men and put an end to their wickedness.
“But Noah found favor in the eyes of the LORD” (Genesis 6:7, 8).
All the people on earth died except Noah and his family who were found righteous in the eyes of God. The people who perished in the flood died because they refused to acknowledge God or seek His forgiveness. Noah, on the other hand, is described as righteous, blameless, and obedient in that he
“walked with God” (Genesis 6:9).
Noah and his family were not spared because they were without sin, but because Noah was obedient and without blame. God had a plan and that plan involved Noah and his family. The polluted, unrighteous population of the world of Noah disappeared from the earth. Mankind was salvaged, and from the line of righteous Noah came the Son of Man, Jesus Christ, who died to atone for the sins of the world. 
Hebrews chapter 11 gives examples of other imperfect people who were used by God in the outworking of his divine purpose. They were all sinners, but God enabled them to overcome and to do incredible things to his glory.
God, in his sovereignty, has the undisputed right to bless and to give, to show mercy and dispense his grace on those whom he favors. God blessed Abraham (Genesis 12:3) because Abraham believed God, and his faith was credited to him as righteousness (Genesis 15:6). Noah was righteous and blameless “and he walked with God” (Genesis 6:9). God showed mercy to Noah because he was obedient and his faith in God never wavered.
Source: The Genesis Flood: The Biblical Record and Its Scientific Implications, 50th Anniversary Edition. (Dr. John C. Whitcomb, Dr. Henry M. Morris
Salvation from sin can only be received as a gift of grace - unmerited favour, not because of anything we have done. Grace is God coming down to move people to places of well-being - at His initiative. That could be physical salvation - as with Noah - giving him plenty advance notice of the coming flood, and instructions for building an ark to save all who would get on to it. "Noah found grace in the eyes of the Lord" (Genesis 6:8). Grace often has something to do with being rescued by God.
The entire Bible consistently shows that it is only by an act of grace that God delivers anybody from anything. Nobody can do anything that entitles them to effectively say to God, “Now look at how high my standards are; I have earned your favour and help!” No, unmerited grace is what gets anybody up and running in God's life and gives a gloriously ‘saved’ future. But our whole life depends on God's grace! Without post-conversion grace, the godly life suffocates. After surviving the flood of water, Noah almost drowned in alcoholic drink! He was not immune from sin and breaking God’s requirements for godly living after experiencing pre-flood grace, yet Noah continued to experience post-flood grace despite his sin.
The lesson for us is that all who would please God depend on the grace of God at every turn. Even if we did everything we ought to do, that would not entitle us to merit from God, as Jesus pointed out in Luke 17:7-10. God has his reasons for choosing anybody even though we may not be privy to those reasons. The Sovereign God does not have to answer to us for his choices or explain matters to us. However, he has explained a great deal in the written record, as in Genesis, so that we find that book full of examples of the grace of God towards sinners. The rest of the Bible continues in that vein. In Genesis, we learn that God was pleased to have Noah “walk with him”, showing that Noah had a heart for God and desired to please his maker, despite his sin. This is of great encouragement to the rest of us, his descendants! We can hope in God, despite our sin, despite not deserving the grace of God. He declared himself to Moses to be “compassionate and gracious, slow to anger, abounding in love and faithfulness, maintaining love to thousands, and forgiving wickedness, rebellion and sin. Yet he does not leave the guilty unpunished.” Exodus 34:5-7 It is the very nature of God that accounts for how he dealt with humanity in general, and Noah in particular, in the book of Genesis.
You could ask this same question of many of those whom God chose to work out His judgments and plans.
Abraham lied, or did not fully tell the truth about Sara being his wife (Gen 20:2).
Jacob lied to his father, Isaac, in order to receive the blessing of the first born (Gen. 27:19).
In anger Moses killed an Egyptian (Ex. 2:12) who was mistreating Moses' brethren.
David committed adultery with Bathsheba, and sent her husband Uriah back to the front of the battle guaranteeing his death (2 Sam. 11:15-17).
There are many other examples that could be listed. Mankind is sinful, as sin originates in the heart of man (Gen. 6:5; Matt. 15:18). But, God always chose those who would listen to Him, who would obey His commandments, and who were at heart faithful to Him.
The one common aspect of all of them is that they were faithful to God.
"And the Lord said unto Noah, Come thou and all thy house into the ark; for thee have I seen righteous before me in this generation." Gen. 7:1 KJV
"By faith Noah, being warned of God of things not seen as yet, moved with fear, prepared an ark to the saving of his house; by the which he condemned the world, and became heir of the righteousness which is by faith." Heb. 11:7 KJV
"And spared not the old world, but saved Noah the eighth person, a preacher of righteousness, bringing in the flood upon the world of the ungodly;" 2 Pet. 2:5 KJV
Abraham was obedient to God, and his obedience was righteousness with God (Rom. 4:3, 9, 13). God called David a man after His own heart (Acts 13:32). God preferred Jacob over his brother Esau (Rom. 9:13).
So it is our faith in God, carried out through belief and obedience to His son Jesus Christ, the seed of Abraham, that provides the opportunity for His grace and forgiveness of our sins (Psa. 130:4; Dan. 9:9; Acts 5:31; 13:38; Eph 1:7; Col. 1:14).
Noah was chosen because of His faith, belief and obedience to God.
Wouldn't the simple answer be, to fix the very problem. Noah walked with God (Genesis 6:9). Walked with God, the same path, in agreement.
Genesis 1:26 reads: 26 Then God said, “Let Us make man in Our image, according to Our likeness; let them have dominion over the fish of the sea, over the birds of the air, and over the cattle, OVER ALL THE EARTH and over every creeping thing that creeps on the earth.”
God gave man control of the Earth with no strings attached. If God wipes out all "man" then he might as well destroy the whole earth, because man has to be the one to redeem it since God gave us Dominion.
Think of it legally. If someone gave you a car, then decided they wanted it back, you could take them to court. The same would apply here, but God knows what he says is set in stone, because he's the judge, and Noah was the best humanity had to offer.