How do Doppler Effect and Time Dilation differ?

One significant difference is that the doppler effect is dependent on the direction of the velocity, while time dilation is only dependent on the speed. This is why the doppler effect changes when A passes B, while the time dilation would be the same before and after. For this reason there is also no doppler effect when something moves perependicularly to you, while there is still time dilation.

Another difference is that the doppler effect is "stronger" than the time dilation effect when something moves towards you or from you. The doppler effect gives a factor $(1\pm\frac{v}{c})$ while the time dilation gives a factor:

$$ \frac{1}{\sqrt{1-\left(\frac{v}{c}\right)^2}} $$

Because of this, the combined effect, the relativistic doppler effect, is dominated by the doppler effect in these cases.

I have given this question much thought. I am providing a link to what I think is the answer, as putting the answer here would be too long.

Please note that I am new to physics so my thinking may be wrong. I would appreciate your comments on whether you think this idea is correct or not.

Thanks!

Link to analysis: https://www.dropbox.com/s/fibfs9uoxjn6lgl/Time%20dilation%20and%20contraction%20effects%20in%20waves%20-%20v48.pdf