My Dad used to own an 88 Hyundai Excel. It had those jet valves that were used in the old carbeurated Mitsubishi engines for emissions purposes. His carboned up over time and stuck. They broke a couple of rocker arms when this happened.

What you are posting is definitely a problem with sticking valves. The head needs to be pulled and rebuilt. Exhaust valves can definitely build up carbon deposits from the soot in the exhaust. It sounds like this engine has been lugged, which would have caused alot of black smoke from unburned fuel. If this engine has an EGR system on it, then it would be entirely conceivable that some of this crud got caked on intakes as well. Regardless, you need a head job on that engine. It would be a good idea to inspect the piston tops as well for impact damage, since these valves are sticking. I am assuming at this juncture, that this engine has accumulated a fair amount of miles as well.