dust comes into my vehicle when the window is closed, looks like it comes from the passenger feet side. there is a peice of fabric that came off where the glove compartment is, could it be coming from there? where else would it come from? there are worn rubber strips along the front passenger door. dust also collects in the trunk; my doors are also started to wear out, can't open the driver side door from the outside and the passenger door won't open unless you open it from the outside and inside. i opened the driver side panel to see what the problem was, couldn't tell, but there is a small metal piece sticking out looking useless where the latch is.
For the dust entry: The fabric you mention is probably insulation over the blower motor. It could be coming in from the outside to that location from the air intake in the cowl near the wipers on the passenger side. It's possible the seal around the cabin air filter is gone. There should be instructions for changing the cabin air filter in your owners manual. You don't actually need to remove the wiper arms if you can get them stopped around 12 o'clock. You could also be getting dust in through the door / trunk seals if they're shot. That should be an easy fix with aftermarket material (Ford parts are probably discontinued).
As for your doors, I've recently been into the driver's door on a Contour. The passenger side is the same, just flipped. There is a cable that runs from the handle on the inside to the latch mechanism. It could be stretched or broken. You need to pull the foam panel to see it. The cable was worn / stretched for a long time on the one I was working on and needed to have the outside and inside handles jiggled to lock the door from the inside before it eventually broke (either that or the mechanism just needed to be lubricated, which I did when I had everything off). The outside handle is connected to the latch via a rod. See Driver side front inside door handle does not work on a 1998 Ford Contour LX. I've got more pictures I've got to upload yet which might help you out. The door control cable is long discontinued (and there are none in dealer inventory or at Vintage Parts), I had to get one from a junkyard. They sometimes show up on Ebay. Looks like the control rod is discontinued, too. If you can find the part number, you might be able to get it from Vintage Parts. This is where I usually take doughnuts down to the local Ford Dealer parts department to get them to look up part numbers. Team Ford might be able to help you out, too.