Radio and seat belt chime do not work?

The chimes come through the radio unit so if it isn't working you won't have the chimes. From the description it sounds like there is a failure in the radio itself. Not uncommon with equipment that isn't OE in an import. Hearing the chime when the fuse goes in sounds like the failure is in the radio itself as the power works long enough to charge a capacitor or activate a chip and then it stops again. Testing it would be to remove the unit and power it up on a bench and see if it works that way. The problem is that you need to look up the wiring for the aftermarket unit, that way you can see if the unit has 2 fuses or not. Many aftermarket units have one for main power and one for battery back-up. You would need to figure out how yours was installed and go from there as installers tend to use less than OE lever methods.

That would be the faceplate. The radio itself would be a box mounted behind it. The idea was that you could pop the faceplate off the radio and thieves would leave it alone. Didn't really work as most people didn't take the face with them when they got out.
I would start by looking up the model of the radio, might have it printed on the faceplate. Then see if you can find the install instructions and wiring for that unit. That would tell you if it has more than one power feed as I suspect it does. If you really want to dig around go to crutchfield. Com and enter your vehicle info and they will show you all the bits needed to install a replacement unit. You could have all the modern things in there with a newer unit. Attached is the wiring for the base radio.

Once you determine the model you have you could post the make/model and I could see if I can find anything that might help.

Okay, it looks like there is a fuse in the back of the unit and that it does have 2 power feeds. (Yellow to a constant power and red to a switched power) Hopefully the issue is that one of those or the ground are the problem.