Each door latch (hatch latches and rear window) has a switch attached to its latch. It's just a simple binary ON/OFF switch that gets triggered when the latch is closed. With age these switches can stick due to wear, dirt and lack of lubrication. Sometimes you can bring them back to life by cleaning the switch buttons and lubricating them with. Unfortunately figuring our which switch is not functioning properly is not easy. The only one that's easy to eliminate is the driver's door, because if it's not working and the key is in the ignition the key chime will go off.
Getting to the switches to clean or replace them is a PITA as each door trim panel must be removed. The switches are attached to each latch with a single Phillips screw.
I've once had a switch that was broken in two and I was able to repair it with JB Weld, but new ones are available and are not very expensive. Often just spraying them down with WD40 and exercising them will revive them.
As far as the door striker bushings, yes they must be in place for the latches to work properly. The HELP section (and even the Ford OE replacement) bushings are crap and don't last very long. I've replaced missing bushings with a piece of 1/2" PEX, which I find makes a very durable repair and a 6' section costs almost nothing and will last you forever. The OD of the 1/2 PEX is a little larger than the OE bushing and you'll have to turn them down just a little bit. I do this by mounting the PEX on the striker bolt, chucking the bolt in a drill (poor man's lathe) and by using a rasp. This gets the PEX down to the OE OD in just about 60 seconds. I use a digital caliper to measure the OD to get it just right. If you want to buy new replacement bushings buy the OE Ford bushings rather than the Dorman HELP stuff.
