Please note that the following is an attempt to explain how some people believe that their opposition to same sex marriage is reasonable and not directed against the homosexual community. It is not an argument for or against same sex marriage, it is merely an exploration of an attitude.
In your question clarification you say:
=> (1) such as the right to have homosexual relationships,
=> (2) public display of affection,
=> (3) the right to constitute a family, adopt, etc,
=> (4) and at the same time say their marriage should be banned
because homosexuality is a sin/abomination. "
From an objective Catholic perspective, as mentioned in a previous answer, everyone is a sinner. Also, from a Catholic perspective, everyone is due respect as a child of God.
As a personal opinion then, I can see that (1) and (2) make sense: respect for the person can be defended and can be advocated in law.
However (3) and (4) are not clearly related to (1) and (2).
I believe point (4) points to an emotional reaction to the issue of gay marriage, and it does so from both sides of the debate. On the anti-gay marriage side this attitude certainly exists and, I beleive, is not particularly helpful in a democracy considering a change to laws. But, on the pro-gay side point (4) points to an equally emotional reaction in that it declares that this is the reason anti-gay marriage advocates oppose gay marriage. So let's be clear: I can agree that many people oppose gay marriage for this reason, but not all opponents oppose gay-marriage for this reason.
A serious, democratically valid, and rational (both independent of religious teaching and within the context of religious teaching) basis exists for many people to oppose gay marriage and it affects point (3) the right to constitute a family, adopt, ...
From a Catholic moral perspective it is legitimate to promote laws that are aimed at reducing discrimination against gays, because discrimination is a moral wrong.
However, these same people can resist gay marriage because the idea of "gay marriage" is incoherent.
To pick at only one thread in this complex, in the Catholic marriage ceremony the couple is required to answer Yes to this question (the wording varies somewhat, this is an approximate quote -- check the exact wording in your country):
"Will you accept children as a gift from God?".
This indicates the root of the more robust opposition to gay marriage.
Again from a Catholic perspective answering "Yes" to this question indicates the following (and this is a very abbriviated list!)
The couple is able, at least in principle, of having children.
People get tied up on this one because the church will marry
infertile couples -- but these marriages do not violate the
male/female fecundity in principle. These marriages still conform to
the principle of male/female/fecundity in their character. So, for
many people marriage and natural procreation through normal and
natural intercourse is a key element of what marriage is. Homesexual
pairings cannot be a "marriage" because they miss this element.
Children are a gift. Any particular couple may or may not receive
the gift, or may think they have received too many gifts, but the
concept "gift" is an essential component of the concept "marriage".
This particular concept is can be almost incomprehensible in
societies like ours where the focus is on the idividual and
individual happiness and above all on control of your life, thus the
widespread use of contraceptives etc. In any case, children are to
accepted and cherished and loved regardless of our own desires in
these affairs. Again, children are essential to the Catholic concept
An essential component of marriage is that is an acceptance of
service to others, especially children. From a Catholic point of
view people marry because each person in the couple wants to marry,
but the commitment extends far beyond the desires of each person in
the couple. Catholic sacramental marriage commits the person to love
the other, for life, whether or not that love is reciprocated. It
also commits the couple to love their children, whether they were
"wanted" or not. The general point here is that a Catholic
Sacramental marriage pretty much ignores the desires and emotional
state of the each member of the couple in the years following the
marriage, even into old age. So, wanting children is not a sufficient
criteria for legitimizing marriage.
There are many other points that can be made along these lines. Most are profoundly at odds with the moderen concept of the person as an independent, self actualized being. Actually these concepts are not really "at odds"; the are radically opposed the many modern ideas of what a person is.
I have presented here a Catholic point of view on some Catholic ideas about marriage. However, similar arguments can be made starting from completely secular theses.
So, part of what worries many Catholics specifically and Christians generally in these debates is that the proposed laws allowing gay marriage actually exclude children as part of the new definition of marriage.
The reference case on same sex marriage in the Supreme Court of Canada is typical of high court decisons all over the world on this issue:
"The natural limits argument can succeed only if its proponents can identify an objective core of meaning which defines what is “natural” in relation to marriage. Absent this, the argument is merely tautological. The only objective core which the interveners before us agree is “natural” to marriage is that it is the voluntary union of two people to the exclusion of all others. Beyond this, views diverge. We are faced with competing opinions on what the natural limits of marriage may be." (from http://scc.lexum.org/decisia-scc-csc/scc-csc/scc-csc/en/item/2196/index.do).
Similar statements, some even more explicit in stating that children have no bearing on marriage, can be found in other decisions.
So, many Christians, non-Christians or even anti-Christians can oppose same sex marriage because they are proundly uncomfortable with the long term implications of a redefinition of marriage that ignores children, nature and the service marriage provides to society. For them the fact that decisions long ago make no mention of children is frivolous: who, before birth control, would have thought it was necessary to mention children? The link was so obvious as to not require any statement at all.
Overall, many worry that the redefinition will, in the long run, be bad for women, children, and for society as whole.