Kernel density plots should be preferred to histograms in nearly all cases. Histograms can be seen as a kernel density plot with a uniform kernel that has been sampled. Since a kernel density plot with a uniform kernel has unbounded frequency content, this sampling introduces aliasing, which is why you get all of these strange effects when adjusting the bin width and offset. In fact, if the distribution of your data happens to be a sine wave, then the histogram will also be a sine wave, but, due to aliasing, it may have a different frequency and phase.
For a kernel density plot with a Gaussian kernel, the kernel size does effect the result, but the situation is much better than with histograms for two reasons:
1. The kernel density plot varies smoothly as the kernel size changes, and so there is greater confidence that you have seen the whole story by only looking at a few kernel sizes.
2. You can construct a kernel density plot with a larger kernel given only a kernel density plot with a smaller kernel. Since the convolutions of two Gaussians produces a new Gaussian with a variance equal to the sum of the input variances, you only have to convolve the small-kernel plot with another Gaussian to produce the large-kernel plot. This, again, means that you have more confidence that you've seen the whole story by looking at only a few kernel sizes.
As a side note, there is technically a 1:1 relationship between 1D datasets and kernel density plots with a Gaussian kernel, and so in theory you don't lose any information by constructing the kernel density plot. In practice, however, you do lose information due to limited precision.
For a kernel density plot with a Gaussian kernel, the kernel size does effect the result, but the situation is much better than with histograms for two reasons:
1. The kernel density plot varies smoothly as the kernel size changes, and so there is greater confidence that you have seen the whole story by only looking at a few kernel sizes.
2. You can construct a kernel density plot with a larger kernel given only a kernel density plot with a smaller kernel. Since the convolutions of two Gaussians produces a new Gaussian with a variance equal to the sum of the input variances, you only have to convolve the small-kernel plot with another Gaussian to produce the large-kernel plot. This, again, means that you have more confidence that you've seen the whole story by looking at only a few kernel sizes.
As a side note, there is technically a 1:1 relationship between 1D datasets and kernel density plots with a Gaussian kernel, and so in theory you don't lose any information by constructing the kernel density plot. In practice, however, you do lose information due to limited precision.