How can I flatten this thin wood back out?
There's a chance that you won't be able to do this to your satisfaction. When sheet goods become warped due to poor storage conditions or bad stacking, or a combination of the two, they tend to take 'a set' which apparently can be impossible to completely remove.
Some improvement is certainly achievable, I've done this successfully myself on warped material, but I don't think you can get thin plywood like this back to absolutely flat which you may require for this application.
It may be possible to flatten plywood by dampening the material, in a high-humidity environment rather than by wetting directly, and then press it flat until it is dry. This works with some other board materials. But the pressure or weight needed would probably be outside the range of what you could rig up as it is in the range of hundreds of kilos per square metre at minimum and possibly a few tons (as you can achieve in an old cast iron book press).
Are there any tricks to storing such wood so that it stays flat?
Storing it horizontally on a dead-flat surface is really most of it as you'd probably guess, but the thicker you stack the more likely minor thickness imperfections in each sheet can accumulate which will mean the upper sheets will no longer be laying absolutely flat.
So shortish stacks on a known flat surface, with some weight distributed on top is probably your best bet. It may also be beneficial to use a sacrificial top board (e.g. melamine-laminated particleboard/chipboard) so that the uppermost sheet of ply is not exposed to the air on one side and not on the other.
I don't know if it applies to 1/8" but with thicker material the usual recommendation in woodworking forums to get more reliably flat stock is to buy better material from a better supplier, although sadly this is still not an absolute guarantee you'll get flat sheets of ply every time.
Also, in case you are unaware "Baltic birch plywood" is by no means one uniform quality to begin with so it's possible you're not getting particularly good stuff anyway, and there is most certainly better-quality plywood than this. If you want to look at getting something high end be prepared for some sticker shock!