Was Jesus opposed to homosexual activity?

Jesus did recall the punishment that had been meted out to Sodom and Gomorrah, mentioning that it was one of the most severe punishments. See Mtt 10: 14-15 :

If anyone will not welcome you or listen to your words, leave that home or town and shake the dust off your feet. Truly I tell you, it will be more bearable for Sodom and Gomorrah on the day of judgment than for that town.

And, why were Sodom and Gomorrah punished by God ? Look at Gen 19:1-5

And there came two angels to Sodom at even; and Lot sat in the gate of Sodom: and Lot seeing them rose up to meet them; and he bowed himself with his face toward the ground; And he said, Behold now, my lords, turn in, I pray you, into your servant's house, and tarry all night, and wash your feet, and ye shall rise up early, and go on your ways. And they said, Nay; but we will abide in the street all night. And he pressed upon them greatly; and they turned in unto him, and entered into his house; and he made them a feast, and did bake unleavened bread, and they did eat. But before they lay down, the men of the city, even the men of Sodom, compassed the house round, both old and young, all the people from every quarter: And they called unto Lot, and said unto him, Where are the men which came in to thee this night? bring them out unto us, that we may know them.

(NB: That quote is from KJV. Many versions have explicit when it comes to Verse 5 )
The aberration of Sodom became so spoken about, that the name became the root for Sodomy, which is a legally punishable offence in many countries.

Yes. Jesus was opposed to any thoughts or actions that run contrary to the intent behind the design of his Father's creation.

Mark 10:6-9 King James Version

Jesus said:

6 But from the beginning of the creation God made them male and female. 7 For this cause shall a man leave his father and mother, and cleave to his wife; 8 And they twain shall be one flesh: so then they are no more twain, but one flesh. 9 What therefore God hath joined together, let not man put asunder.

Jesus never changed any O.T. laws. What he did was to show up pharisaical additional man-made 'rules' that made keeping God's laws a great burden for the people. He showed the real depth of spiritual meaning behind God's laws. For example, he said that a man thinking about having sex with a woman not his wife was committing adultery in his heart. Jesus raised the bar far higher by exposing the heart's desires and hatreds.

Jesus never mentioned lots of O.T. aspects of keeping God's laws. For example, the scriptures never mentioned abortion (as opposed to miscarriage, which is another matter entirely). Yet all Israel knew that to deliberately take the life of an unborn child, or to kill a new-born baby, violated God's law on the sanctity of human life. Does that mean that the lack of any God-given law on abortion means it's 'neutral'? Of course not!

That's why the first century church produced a document called The Didache which states,

"You shall not kill the foetus by abortion, or destroy the infant already born." That was because they believed the biblical statement that God created humans in His image. The Image of God, Personhood and the Embryo, Calum MacKellar pp 8-10 (SCM Press 2017)

So with h*m*sexuality in the N.T. Jesus does not mention it. Paul and John do. Yet everything said in the N.T. about it agrees with all that is mentioned in the O.T. God's mandate to humanity is to fill the earth with offspring from the sexual union of a man with a woman, who were created in his image. Satan is out to corrupt that image in humanity, and violation of God's wonderful gift of sex is one of his main ways of doing that. The extent to which most societies have now gone in accepting ungodly views of sex would have shocked first century Christians to the core, had they known what would transpire in our day. Imagine them reading this article, published in 2016:

"Over a century ago, G.K. Chesterton pointed out that once we lost our faith in God, we would lose our humanity as well. This is happening in front of our eyes. Society is now guided by personal opinions and feelings, so that what is right for you is okay and nobody ought to criticise you for that. Satisfying self is the goal of billions of people, who no longer live to honour God or to put other people before themselves.

This is seen especially in the area of family life and human sexuality. Having got same-sex ‘marriage’ in 2014, our reshapers of humanity moved on to a new shibboleth issue; transgender. We are being bombarded with information that if a 5 year old boy feels that he is a girl, then, yes, he is a girl. Having got h*m*sexual rights accepted on the basis of the doctrine that ‘people are born that way’, we are now being told that we must accept ‘gender fluidity’. Whereas God made us either male or female, Facebook tells us that there are 26 different genders and we can be whatever we wish to be...

Education is now the domain of social engineering. Having rejected Christianity as the foundation for our laws and morality, those in control of society are seeking to replace it with their own laws. Those laws include ostracising those who choose to stay with God’s laws and who will not be intimidated into keeping quiet about the immense danger our children are now in, because of modern brainwashing techniques used upon them." Article by David Robertson, Moderator, Free Church of Scotland, The Record magazine pp 5-6 January 2016

I do not know your age, but I do know that most Christians of my generation in this part of the world were taught in 1960s schools the biological facts about sexual reproduction, without a word suggesting we indulge in any of the sexual libertarianism today's children are being actively encouraged to experiment with. Perhaps you could consider what influences might be at back of you asking this question? You certainly won't find any such influences in either the O.T. or the N.T. On the contrary, the Greek word p**nea in the N.T. covers all sorts of sexual breaking of God's laws, not just fornication or adultery or other sexual matters, but thinking p**nographic thoughts. The massive flood of p**nography in our world today is a major influence dominating peoples' thinking about sex and h*m*sexuality. Satan is having a field-day corrupting minds regarding sex, and never more-so than with young, easily-impressed minds.

The answer to your question is "Yes, he was". Further, all God's laws against h*m*sexuality are opposed to its practice.

Jesus did not directly address h*m*sexual activity in the Gospels so a degree of speculation is necessary here. But he was a Jew who upheld Jewish law on matters of heterosexual activity - and even condemned extra-marital heterosexual fantasies -- so that might be a good starting point:

You have heard that it was said, ‘You shall not commit adultery.’ But I say to you that everyone who looks at a woman with lust has already committed adultery with her in his heart. If your right eye causes you to sin, tear it out and throw it away; it is better for you to lose one of your members than for your whole body to be thrown into hell. And if your right hand causes you to sin, cut it off and throw it away; it is better for you to lose one of your members than for your whole body to go into hell (Mt. 4:27-30)

It might be argued that this applies only to fantasies by married men, but if we take seriously what Jesus said earlier in the chapter, we cannot suppose he would have condoned h*m*sexual behavior:

Do not think that I have come to abolish the Law or the Prophets; I have come not to abolish but to fulfill. For truly I tell you, until heaven and earth pass away, not one letter, not one stroke of a letter, will pass from the law until all is accomplished. Therefore, whoever breaks one of the least of these commandments and teaches others to do the same will be called least in the kingdom of heaven, but whoever does them and teaches them will be called great in the kingdom of heaven. (17-19)

As the OP admits, Jewish law was very strict in its prohibition of h*m*sexual activity:

If a man has sexual relations with a man as one does with a woman, both of them have done what is detestable. They are to be put to death; their blood will be on their own heads. (NIV Lev. 20:13)

It can be argued that Jesus opposed capital punishment for such behavior based on the story of the Woman Taken in Adultery, but Jesus nowhere says that the law against male h*m*sexual activity should be altered.

What are the Christian rules?:

The OP says"

Suppose one was interested in living life only according to what Jesus said and did. Not the scripture and prophets that came before. Not the saints and missionaries that came after. In that case are there any rules on h*m*sexuality? Is it good bad or neutral?

This supposition implies a willingness to ignore both OT and NT proscriptions against h*m*sexual activity except for the sayings of Jesus. It is certainly possible to dismiss Jesus' attitude toward the OT Law in Mt. 5, since many other Christians do likewise based on more liberal attitudes Jesus shows elsewhere. The trouble here is that the best arguments for dismissing OT Law are found in the Letters of Paul, which the OP does not wish to consider.

However, there are many churches who welcome practicing h*m*sexuals and even celebrate gay marriages. Catholic, Orthodox and most Evangelical churches do not condone h*m*sexual activity, but some Protestant denominations do not have rules against it. Here is a resource page that will help the reader identify which churches have rules against h*m*sexual activities and which do not.

Jesus re-iterated the words of Adam, the first man, when he is recorded by both Mark and Matthew saying :

For this cause shall a man leave his father and mother, and cleave to his wife; [Mark 10:7 KJV]

For this cause shall a man leave father and mother, and shall cleave to his wife: and they twain shall be one flesh ... [Matthew 19:5 KJV]

Therefore shall a man leave his father and his mother, and shall cleave unto his wife: and they shall be one flesh. [Genesis 2:24 KJV]

Jesus, therefore, reiterates and supports the words of Adam regarding the purpose and the function of male and female in regard to personal and physical union.

No other alternatives are ever considered by either Adam or Jesus.

If Paul was speaking as an authorized apostle and representative of Jesus Christ (see Romans 1:1), as most Christians hold, then Paul's teachings reflect the mind of the Lord on the matter.

Many Bibilical teachings have been rejected or reinterpreted by modern audiences because they do not align with contemporary social priorities - but it would be anachronistic to superimpose those contemporary social priorities upon the intent of ancient writers.

If one rejects the portion of God's teachings that are uncomfortable, one is left with a god created in man's image rather than man created in God's.

Paul did not specifically target one group only in his calls for repentance - he taught that we are all sinners (see Romans 3:23); the glorious message of this epistle is that reconciliation with God is possible: we have all fallen in sin, but we are not trapped there.