Will a "700 x 25-32cc" inner tube fit my "700Cx28/38c" / "ETRTO 622x20" rim?
It looks as though you're confused about the range of allowed tyres on the rim vs the tube. What connects the two is the size of the tyre you actually have.

There are two independent ranges. The rim is rated to take tyres in the range 28 to 38mm wide. Your tyre will be an exact size, and ideally should be within that range. You can measure the width or guess, because it's not critical.

That's because the tube will be rated to fit a range of tyre sizes. The one you're looking at will fit any time from 25mm to 32mm wide. 

As long as the tyre you actually have is more or less the size of the tube, you'll be fine.

In practice a thinner tyre will be fine, it might just be hard to stuff all the tube into the tyre when fitting it (it's more obvious if you're trying to fit a 55mm wide tube into a 22mm wide tyre... and that example might not actually work because there will be so much rubber filling the tyre that you might not be able to get any air in).

The same applies to tyres, BTW. You could probably run 25mm or even 23mm tyres on that rim without problems, and likely 40mm or 45mm wide as well, if the tyre cleared your frame. I dont suggest you actually do run wildly out of spec tyres, but people do (I have run 55mm tyres on narrow rimes before).

Also, tyres fit a range of rim widths :) One guy I met ran narrow tyres on wide rims because he was convinced it lowered the rolling resistance. He may have been right, but it made hard cornering quite exciting when the rims hit the ground.

I suspect you're over-thinking this. It's straightforward once you understand it. (rim takes a range of tyres) + (tyre is one exact size size) + (tube fits a range of tyres).