Is what you are asking: Given two equations for ellipses, is there a way to construct a third equation that only represents the intersection/overlap of the two ellipses?

No. Not the intersection; but rather the entire combined area. Without the first notion of how that might be possible.

I'm a mathematician in the same way that I'm a plumber or electrician or car mechanic; I generally have the ability to lookup enough information to allow me to solve my immediate problem; but ask me about it a month or so from now and I probably can't explain how I did it.

Years ago I was asked to give a financial type help with a spreadsheet he was developing for shares. He wanted to isolate the intersections/overlaps between 3 or more spot price curves. He had an line-fitting algorithm that approximated a given curve by producing some kind of multi-term polynomial. The first term approximated the line crudely; the second term refined it; the third refined the results of the second and so on. His task was to combine the polynomials from the 3 or more spot-price curves into a single polynomial, and then iterate it between predefined limits. It was complex stuff that I won't pretend to understand.

However, what I was able to do was help him redefine his formulae, extracting common sub-terms and the like, that resulted in a 8 times reduction in his spreadsheet run times (measured in hours). Then, when he'd proven that what he was doing had some merit, I coded up a program in C to do the same calculations, that cut that time by almost another order of magnitude. I didn't have to understand the formulae; just be able to code up the parts and execute them in the right order to produce the same output for the given inputs.

From 4+ hours to just over 3 minutes. Still too slow to inform spot price dealers; but with today's hardware they are probably doing the same stuff millisecond by millisecond in automated traders.

And it was with that in mind that I thought that there might be some way to pre-process the equations of a bunch of ellipses into a polynomial whereby the first term would put (say)90% of points in or out; then the second term 90% of what's left; and so on. With short circuiting, that might be faster than testing every point against every ellipse individually.

I still think there is some possibility of deriving the polynomial; but I am simply not equipped with the tuits to see how to even begin such a thing.

Similarly for the two ancillary questions at the bottom of the OP.

I can, by projection of stuff I've seen done in other fields (maps and the like), see that both are probably feasible -- if a blind mathematician can turn a sphere inside-out without breaking the surface (topologically speaking); then mapping a multi-segment curve to a straight line is probably child's play. If you're the right child. I'm not. (Either a child or the right one :)