How can I change the default double-mouse-1 word selection behavior so that I can move the point around right afterwards to extend/shrink the region? By default, if I forward/backward-char after selecting the word by double-mouse-1, it moves the point, but the mark disappears, so nothing is selected.
You don't need to change anything.
You just need to click
mouse-3 where you want to put the new region limit (whether enlarging or shrinking, either end, as many times as you like).
mouse1to select a word or symbol etc. (Or triple-click to select a line, etc.)
mouse-3to extend or retract either end of the region.
Repeat #2 as often as you like, to adjust the region.