I've just been on the AA website. I was planning on going on a drive through the Alpine passes, but now I'm worried.
1) What is the best time of year?
From May to September. You could go in April but there would be a risk of snow on the higher passes. It is always possible, but moderately unusual, to get snow even in summer. I would tend to avoid July and August as they are the peak summer holiday season.
2) Snow chains are compulsory on certain Swiss roads. How will I know which ones? How will I fit snow chains?
They are only compulsory if there is snow. There is not an obligation to carry chains but there is an obligation to have them fitted when the sign is displayed and there is snow. My advice would be not to take a high pass if snow is forecast. There is an obligation to carry a first aid kit and triangles in some countries.
3) What happens if I break down and I can't speak enough French, German, or Italian?
Don't worry you will have Assistance Insurance won't you? It is nearly always possible to find someone who speaks English. People that work at hotel reception desks or restaurants usually have at least some English.
If you come this way send me a PM.