The house passage would still stand. All you need to do is pass the same bill without change, and it would go to the President. It doesn't necessarily die because this iteration failed.
This is correct; when one house passes a bill and it is passed with amendments by the other, the original house has two options: either to pass the amended version (in which case it goes to the president) or to reject the amendments. In the latter case, the second house then holds an immediate up-or-down vote on the original version of the bill which, if passed, goes to the president.
Yes, but that is not what happened here. This was a House originated bill, The Senate voted down that House version unaltered. What you are saying would apply if this had been amended and passed by the Senate and then sent back to the House, where it then failed.
The Constitution states that in order for a bill to become law it must pass both houses in the same form and be signed by the President. Nothing in the Constitution would prohibit the Senate, to my knowledge, from considering and a passing a Senate bill worded the exact same way as the House bill. Since the same text would have at that point been passed by both chambers, it would then go to the President and that was my recommendation for how to resolve this situation.