Apollo Ventura 4000 started smoking, and the plastic around the mirror melted slightly, as well as on the top case above the bulb. What could've gone wrong, and how can I fix it?

probably an electrolytic capacitor that has failed
white smoke ?

Brand and model would help but I'll guess at this. If micro is a wall hung unit over the range, you usually need to remove the top grill assembly above the door. Then look inside directly above the area where the bulb is located. You should see a metal cover with usually 1 screw in it to remove. Once screw is out that metal plate usually pulls forward and upward to reveal the bulb and socket. Replace bulb and reverse procedure to assemble.
If the micro is a counter top model---the complete outer case usually needs removed if you don't see a small access door by the bulb. Remove all screws around back edge and any screws in the side panels---then lift back edge and pull outer case rearward to remove.

Remove the top grill assembly. It is usually held on by 2 screws going down through the cabinet top. You may need to tap the grill to the left slightly to get the tabs to release. Once the grill is off, look near the middle of the top edge of the microwave frame for 1 screw that looks like it goes through a metal lip. It will be right above the inside light bulb area. remove screw and then pull the metal tab towards you. The complete light bulb assembly will slide up and out towards you. Replace bulb and reassemble microwave.

Sounds like it over heated and you have some wires that might have melted and are causing this issue. In either case I would not recommend using it any more for fire hazards and would recommend you replace it. repairing these is not economical. You can unplug it and let it set over night then remove the case and see if you can find what has gone bad and decide if you need further info on a repair if you chose that route but i can't tell you what the problem is without seeing it or a better description of what is burnt.

Plastic will melt but the door frame is metal and will not. So there is no safety issue here, just looks.
One could replace the door assembly if required. I would leave the door open after convection cooking to let the heat escape without heat build up causing the plastic to mishape.

Not sure on your model but my Kenmore Microwave has the bulb up above the plastic ceiling inside.
It's semi-rigid so you flex it slightly and the tangs come free and it comes down.

Mr. Muscle Kitchen cleaner will remove the smoke. It won't hurt the plastic. Just make sure to either wear rubber gloves or wash your hands soon after cleaning off the smoke residue. It will make the skin on your hands / fingers slick and then dry after your done. Let the cleaner sit for about a minute before trying to wipe off the smoke residue.

I have a microwave above my stove also but it has not done what yours is doing. I would try a lower watt bulb since it would not give off as much heat. You may need to replace the lens or else just take it out and get rid of the problem altogether. You would have a brighter light anywas even if you kept the same bulb. Most of those bulbs are florescent and do not put out much heat anyway and seldom cause heat problems, esp. giving off enough heat to melt something.

Advantum 120 over range microwave - with burned out interior bulb: Remove top fron grill cover -- held byu 2-3 screws on top. Remove sheetmetal plate that lies parallel to top of unit, just to the right of the square plastic porting on the far left of the unit. The sheet metal plating is about 12" wide and 8" deep and is shaped like 2 overlapping rectangles. It is held in place by 1-2 screws. (I don't recall if you needed to remove other plastic grid on right front -- I did, but I was still searching for the bulb). The bulb is located about 5" from the left side and about 2" in from the front, covered by a small sheetmetal box. THe bulb looks like the little halogen bulbs used in low-voltage outdoor lights - but is slightly larger - DO NOT REPLACE WITH THAT KIND OF BULB). Remove the one screw holding the box, unplug the bulb and find a replacement. Can't tell you where to get a bulb -- that's what I'm doing next. Good luck.

On most microwavre ovens all you have to do is open the door all the way and lift straight up at the hinges. The door comes right off the hinge pins. In some cases you may have to remove the top louver panel (right above the door). You should NOT have to unmount the oven to complete this. In fact...I would not recommend it.

Is this a simple matter of replacing a door handle? If you need to only replace the door handle, you just lift slightly on the INNER door plastic around the glass and you will be able to access the door handle screws. There are a series of small grooves in the panel that you can use a small screwdriver to gently pry. You should not have to remove the entire panel. Just open it enough directly behind the door handle. Be careful when prying on the plastic. It can break if you're not careful. I hope this helps you.