Lovely -- I started to draft a reply but got bogged down in detail. You've said it in a nutshell.
Balanced forces are balanced momentum flows. As with flows of mass or charge, you can have flows occurring without any net accumulation or depletion in any body.
As for the strain - it's essentially Hooke's Law. The (directed) rate of momentum flow by the Coulomb interaction between molecules depends on their separation. Their equilibrium separation (strain) therefore depends on the (directed) rate of momentum flow they are required to sustain (tension or compression).
Thanks K^2 for your earlier post, too -- yet more impetus for me to get around to learning about symmetry in physics. Nice to know I'm intuitively on the right track (if only in a tiny way) Any suggestions for introductory books or materials, suited to a time-poor person, would be gratefully received!