How does one practically test all solder connections on a circuit board?
Give it a visual inspection with lots of light, and use a head-mounted magnifier or a bino microscope if your eyes are not perfect. Get light from the sides as well as head on.
You can inspect a couple hundred joints in a minute or two with experience.
Knowing what to look for is the trick- you're looking for joints that have the lead with a hair around it showing they've broken free or pads that have a hair mark to the trace showing the pad has lifted. On dense SMT boards look for joints that are not smooth and rounded with an appropriate amount of solder.
Pay particular attention to heavy components, components that interact with the user's ham-fisted motions and around fixing points and connectors.
I would not suggest blindly re-soldering connections, you're more likely to cause problems than to fix them and you'll make it impossible to repair later.