As to the first part of your post you need to be specific as generalities are usually wrong. The Bible has been demonstrated to be historically accurate by archeologists.
However lets deal with the one verses you posted, but lets put those verses in there full context.
Matthew 15:1-20
15:1 Then some Pharisees and scribes came to Jesus from Jerusalem, saying, 2 "Why do Your disciples transgress the tradition of the elders? For they do not wash their hands when they eat bread." 3 And He answered and said to them, "And why do you yourselves transgress the commandment of God for the sake of your tradition? 4 "For God said, 'Honor your father and mother,' and, 'He who speaks evil of father or mother, let him be put to death.' 5 "But you say, 'Whoever shall say to his father or mother," Anything of mine you might have been helped by has been given to God, " 6 he is not to honor his father or his mother.' And thus you invalidated the word of God for the sake of your tradition. 7 "You hypocrites, rightly did Isaiah prophesy of you, saying,
8'This people honors Me with their lips,
But their heart is far away from Me.
9'But in vain do they worship Me,
Teaching as doctrines the precepts of men.' "
10 And after He called the multitude to Him, He said to them, "Hear, and understand. 11 "Not what enters into the mouth defiles the man, but what proceeds out of the mouth, this defiles the man." 12 Then the disciples came and said to Him, "Do You know that the Pharisees were offended when they heard this statement?" 13 But He answered and said, "Every plant which My heavenly Father did not plant shall be rooted up. 14 "Let them alone; they are blind guides of the blind. And if a blind man guides a blind man, both will fall into a pit." 15 And Peter answered and said to Him, "Explain the parable to us." 16 And He said, "Are you still lacking in understanding also? 17 "Do you not understand that everything that goes into the mouth passes into the stomach, and is eliminated? 18 "But the things that proceed out of the mouth come from the heart, and those defile the man. 19 "For out of the heart come evil thoughts, murders, adulteries, fornications, thefts, false witness, slanders. 20 "These are the things which defile the man; but to eat with unwashed hands does not defile the man." NAS
Jesus is addressing the hypocrisy of the "traditions" of the elders; Jesus is not contradicting the Law.
Now if you would care to post those "contradictions" you find in Scripture we can discuss them.
As to your second point.
Sodom and Gomorrah were not destroyed because of homosexuality. So that example is fallacious to begin with. Next, Jesus' earthly ministry was to the Jews and Jesus did not contravene the Torah or the prophets. If you know anything about Christianity then you should understand that gentile (non-Jews) Christians are not subject to the Laws in the Torah because of Acts 15:
Acts 15:22-29
22 Then it seemed good to the apostles and the elders, with the whole church, to choose men from among them to send to Antioch with Paul and Barnabas-- Judas called Barsabbas, and Silas, leading men among the brethren, 23 and they sent this letter by them,
"The apostles and the brethren who are elders, to the brethren in Antioch and Syria and Cilicia who are from the Gentiles, greetings.
24 "Since we have heard that some of our number to whom we gave no instruction have disturbed you with their words, unsettling your souls, 25 it seemed good to us, having become of one mind, to select men to send to you with our beloved Barnabas and Paul, 26 men who have risked their lives for the name of our Lord Jesus Christ.
27 "Therefore we have sent Judas and Silas, who themselves will also report the same things by word of mouth.
28 "For it seemed good to the Holy Spirit and to us to lay upon you no greater burden than these essentials: 29 that you abstain from things sacrificed to idols and from blood and from things strangled and from fornication; if you keep yourselves free from such things, you will do well. Farewell. " NAS
The broad definition of "fornication" is sexual immorality, which includes incest, adultery, bestiality, and homosexuality.
As to what Jesus did nor did not address during His earthly ministry that is an argument from silence and is a logical fallacy. The fact Paul lists same gender sexual relationships as sin is covered by what Jesus taught concerning living in obedience to God and the decision of the Council of Jerusalem.
Now why is Paul the authority? Paul was called specifically to be the apostle to the gentiles and all his letters except Hebrews were written primarily to non-Jews.
Jesus did not judge the woman taken in adulter and did say:
John 8:7
"He who is without sin among you, let him be the first to throw a stone at her." NAS
However, He also said to the woman:
John 8:10-11
10 And straightening up, Jesus said to her, "Woman, where are they? Did no one condemn you?" 11 And she said, "No one, Lord." And Jesus said, "Neither do I condemn you; go your way. From now on sin no more."] NAS
Chrisitans should not judge non-Christians.
1 Corinthians 5:9-13
9 I wrote you in my letter not to associate with immoral people; 10 I did not at all mean with the immoral people of this world, or with the covetous and swindlers, or with idolaters; for then you would have to go out of the world. 11 But actually, I wrote to you not to associate with any so-called brother if he should be an immoral person, or covetous, or an idolater, or a reviler, or a drunkard, or a swindler-- not even to eat with such a one. 12 For what have I to do with judging outsiders? Do you not judge those who are within the church? 13 But those who are outside, God judges. Remove the wicked man from among yourselves. NAS
However, this does not mean we cannot "judge" a behavior based on Biblical truth. And what this does tell us is that any professed Christian who is actively gay is fair game for being judged.
Did I address your points?