How does analog radar signals encode two dimensions

As you've mentioned, all radar works by measuring the TOA (Time of Arrival) of a return pulse that has reflected back from an object. However, SAR uses a lot of special signal processing to compensate for the fact that the actual antenna is much smaller than the synthetic aperture.

The movement of the physical antenna from the time it emits an incident pulse to the time it receives a reflected pulse is what makes the synthetic aperture. So, among other things, the doppler effect of that movement on the frequency of the pulse and the fact that different frequency components of the incident pulse get reflected back to varying degrees (group delay) are all measured and compensated.

In addition, modern SAR is focused rather than unfocused. That is, similar to lenses in optics, phase adjustments of components of the radar return can be processed to effectively focus the pattern and increase the resolution.

I'm no expert by any means, but those are the basics. If you want to get a little deeper into it (with all kinds of lovely calculus equations), you might enjoy reading this book chapter specifically on that topic. That document is a couple of decades old, but the principles are the same.

If a picture is worth a thousand words, as the old saying goes, http://www.intro2radar.com/ is worth many, many thousands because it has not only pictures but animations. You may find the "ranging imaging" animation particularly helpful to answer your question. It's also worth pointing out that the animations are intended to complement the excellent book, Introduction to Microwave Remote Sensing by Dr. Iain H. Woodhouse.

To answer your question: No, you can't generate your local oscillator that way.

The output of the first mixer will have both sum and difference frequencies in it. In other words, in addition to a signal that runs from 2201 to 2601 MHz, you'll also have a signal that runs from 2199 to 2599 MHz, and there's no way to eliminate just one of those signals.

It isn't clear to me why you want an additional frequency offset anyway. The usual idea with this kind of radar is that the reflected signal already has a frequency offset from the current transmitter frequency that's based on the round-trip delay between the radar and the reflecting object. The beat frequency gives you a direct measurement of the distance to the object. And if it has a radial velocity, by observing how the beat frequency shifts when the transmit frequency switches between rising and falling, you can measure that as well.