Let's say that the New Testament remained pretty much the same as it is today, but with one major exception: There was no reward after you die, nor was there a punishment for those who do not follow Jesus' teachings. All of Christianity's potential benefits must take place during your lifetime alone.

A related question, to all Christians this time, would be: Would you still be a Christian even if God didn't offer you a seat in Heaven?

Let's turn this around.
Would it be as successful if there was no punishment of eternal damnation in hell?
More care for the punishment than for the reward

I think maybe not... it's an important part of Christianity. One reason Christianity was so successful is that it was universal, you didn't need status or anything. Anyone could get into Heaven. That and its message was given to anyone and everyone that was willing to accept it, including low status men.

In other words, loving your enemy and offering shelter for the poor would not be preferable if it was not for the eternal reward, and the word of God isn't good enough on its own?

Sounds like God applies to your greed more than anything, if Heaven has such a big role in Christianity's spreading.

Of course not. Why do you think they invented heaven and hell if not for recruiting purposes.

Now Mohammed really kicked things off with the offer of virgins in the afterlife.

You should also mention the hell punishment. Obviously everybody wouldn't follow a religion if there was no reward and/ or punishment.

big problem with the initial question, Heaven and hell are answers to a question and not just a reward/punishment. MANY people need to know(or think they know) what will happen to them after they die.

how can any religion be successful if it doesnt answer that question.

I mentioned it pretty clearly in the OP. Just didn't want to make the title too long.

And Heaven/Hell is not the only reward/punishment that religion has to offer. Is it?

I think one of the reasons Christianity is so appealing is because it gives people this incentive; Heaven. You know, what's in it for me if I believe in this Jesus guy? Oh, I get eternal life in Heaven, well that sounds pretty good. It appeals to our selfish nature. 

Heaven wasn't meant to be used as a means to control people, but that's what it's come to. 

I'm saying if there wasn't Heaven it wouldn't be the same religion, I do not think it would be as appealing in general. In a sense, Heaven can be used as an incentive for morality. If you do or do not do things because you are afraid of going to Hell, that is the wrong reason. 

Let's be clear that I did not say that my religion is dependent upon me going to Heaven. I was merely answering the original question. But I think Christians would behave differently if they did not think Heaven existed. The idea of Heaven appeals to our greedy feelings. Having this list of criteria you must meet in order to go into Heaven seems to me misleading and wrong, because it can make people's moral actions derived from fear of going to Hell, which is the wrong reason to behave morally. Doing the right thing in order to get into Heaven is also the wrong reason for doing the right thing. What we should be doing is the right thing regardless of whether or not Heaven exists, but I fear that the majority of people do not do that.

You can't have Christianity without heaven. It's not a possibility. The question itself is an impossibility, so there's no reason to ask it, because it can't be answered.