You have yet to show how morality could evolve where exactly on the chromosome the morality gene is carried.
Nobody is claiming there is any such thing as a 'morality gene'. The structure of our brains and the various organs which produce hormones and other mood-altering chemicals is dependent on a whole host of genes. Taken together, they mean that human thought works in certain ways - leading us to feel compassion for and empathy with others; and to develop concepts of fairness.
And if it is a genetic in nature than why does it seem to vary greatly from culture to culture? Certain cultures have endorsed killing while others punish killing.
All cultures punish killing, and most endorse it. What changes is the specific circumstances under which killing is acceptable, praiseworthy or abhorrent. The reason for this change is because we don't have a specific set of commandments hardwired into our genes. Rather, our genes determine that we will possess certain traits like compassion, guilt and a sense of fairness. It's from these that people construct moralities, and the specifics are bound to be dependent on the cultural situation.
This claim is not well grounded; I think whether an individual will be rewarded for his altruism will depend on the place and time, and the overall surrounding circumstances. For example if you die during self-defense how can you be repaid for this act? Or if you take care of your pet whose going to reward for taking care of your pet. If you were going to crush a bug underfoot but noticed it just on time who will honor you for that act of mercy.
If we're talking about the genetic basis of morality, then it's irrelevant whether an action will actually benefit you (or your genes, to be more precise). What's important is whether, on average over the course of our evolutionary history, the drives which caused you to behave in that way brought about behaviour which was beneficial for your (ABE: ancestor's) genes.
That sentence was long and confusing, so I'll try anf elaborate. Imagine a much more simple animal, without conscious thought. It works on a few simple instructions to decide its behaviour, as these generally work out well for it and lead to reproductive success. It lives underwater, and needs to come to the surface to feed in the daytime. The simple instruction that causes it to do so is to react to light by swimming towards it, and this all works well.
However, imagine somebody builds an artificial light over the creature's pond. Reacting to the light, it swims to the surface, wasting energy thrashing around for a food source that isn't present and with daytime camouflage causing it to stand out to night time predators. Despite this behaviour being wasteful and dangerous, we wouldn't conclude it was intentionally committed suicide or working to its own detriment. We'd know that it was following rules that are generally beneficial, but in this particular case backfired.
In the same way, if someone behaves in a fashion which disadvantages them, and disadvantages their genetic legacy, we can't just conclude there can be no adaptive reason for this behaviour. They'll just be working on cues which have lead to genetic success for its ancestors in the past, but in this particular instance backfired.
Why should we care about the survival of others?
Throughout most of the history of humanity, we lived in small groups of closely related people. Most people you saw regularly would be close relatives who share similar genes to you. If you have genes which predispose you to help your friends and family, even to your own detriment; they will probably share the same genes. So, anything you do which aids your family's survival and success will lead to the propagation of these 'caring genes' - even if you yourself die or miss out in the process.
It's not about consciously caring. It's about being inclined towards behaviours and attitudes that have lead to genetic success for our ancestors.
Edited by caffeine, : No reason given.