Hdtv converter box on older tvs?

Yes, it should (usually). Depending on the outputs and options on the box by brand and cable/satellite provider typically it will work. However you usually pay more upfront or per month on HDTV boxes, so unless it's hooked up to a HDTV you don't need an HDTV box and are paying more for nothing.

Sometimes if the box is set to output HDTV at 1080p, 1080i, 720p or 480p, you might have problems hooking it up to your TV and seeing a signal, which allows you to change settings on the box. In this case, use a pin and reset the box which typically will put the box in a low res mode so it can be hooked up to a standard signal. IF the box is outputing a HDTV signal and you have it hooked up to a non-HD television - you won't be able to see the signal. So just because it's hooked up right does not mean you will see the signal. For HDTV televisions they can detect this and show the signal 99% of the time.

Yes. A 1988 tv will work with a converter box to receive the new digital signals. You'll just need to pick up one of those new fancy digital receiver boxes to do so, and then some extra gadget maybe to hook it up to your tv.