Hannukah is a Jewish holiday celebrated at the ed of the year. So, it is not actually an English word but a Hebrew word. Its most famous symbol is the Menorah, a beautiful candleholder with 6 or more candles.

You would say "I have a stomachache/headache". I've never heard anyone say "earache". I would say "My ear hurts" / "my ears hurt". You can also say "my stomach hurts." "I'm sick" can mean "I am ill" or "I am nauseous". You can also say "I am feeling sick." "I'm not feeling well" also works for all of these. - I'm feeling earache/stomachache - I'm with an earache/stomachache - I'm sick (could it have the same meaning as "I'm nauseous" or just to talk about diseases?) - I'm with a terrible headache

Because it's saying "car OR PLANE" I think it's referring mostly to long distance travel. It's probably encouraging people to take the train, which are more environmentally responsible, or to not travel to far places as much.