I'm not sure I'd go so far as to call it hate. I encounter many of them with a sense of foreboding, though.
Your statements will do nicely to illustrate.
"We really believe in what we teach."
That I accept, but the catch comes with the next bit.
"Most Christians are trying to help you by telling you the truth, and most non-christians think we are trying to push it down their throats."
Well we would, if you phrase it like that!
I'm not sure that you would like the reverse:
"I'm just trying to help you realise the truth that there is no god."
Try "Telling you what we see as the truth..."
and note how different it can feel.
"Do you not see that we believe with all our hearts..."
Yes, but sincerity alone has little use here.
Depth of conviction does not correlate to truth.
(Otherwise the most dedicated suicide bomber or other fanatic would be right by default)
And sincerity (ok, and lack of it) can be found on any side of an argument.
"Have you ever looked at this from our side, instead of hating us for sincerely trying to help you?"
Yes, most certainly. I can see where you are coming from. In most cases you (generic "you") are being consistent with your beliefs.
But have you? I have my beliefs, which I hold because I think they are true, (approximately, corresponding to reality). If a Christian is not prepared to consider in debate the possibility that I am right, and that their view is the one in need of correction, why should I be inclined to listen, having had my views effectively dismissed as worthless from the outset?
If understanding for beliefs is being requested, should it not be reciprocal?
"If a child was running into traffic, would you only tell them once, or do everything you could to save their lives?"
Yes, but is that the situation? An illustrative parable can be set up to argue any point of view.
The "all religions are the same" one is quite popular, describing people climbing routes up different faces of the same mountain.
And no, I don't believe that one's true either.