If you have exact zeros in your data, the distribution they were drawn from will not be normal. So rather than remove them, they're grounds for immediately rejecting the null of bivariate normality.
Elliptical contours imply an elliptical distribution. A bivariate normal is elliptical, but so (for example) is a bivariate $t$. However, elliptical contours with normal margins would imply bivariate normality. So that part would be fine, if you knew that information about the distributions from which the data were drawn.
However sample margins that aren't obviously non-normal and sample contours that are not obviously non-elliptical doesn't mean that the distribution is bivariate normal. Similarly failure to reject in some hypothesis test of such things doesn't tell us that the distribution
the data were drawn from is bivariate normal. Sometimes you can see
that you don't have bivariate normality and sometimes you can't tell
that you don't.