When is a cake ready?

is an oven thermometer the same as a meat thermometer or a sugar thermometer.

No, it's a thermometer that lives in your oven and tells you what the temperature really is. Here is one in action.

A meat thermometer is stabbed into a hunk of meat to assess internal temperature. What you are calling a "sugar thermometer" is probably what I'd call a candy thermometer. It sits inside a pot & measures the temperature of whatever is cooking--candy, or jam, for instance. A lot of them double as deep fat frying thermometers so you can keep track of the oil temperature.

There are a number of thermometers that can do all those jobs--meat, candy, and oil.

Yes, next time, dry the carrots.

When is a cake ready?

Ye bring out the philosopher in me, OP: when it is ready.

When the carrots are dry, when the eggs are the right size, when the thermometer is in the right place, when the toothpick is sort of moistish but not sticky, when the tin is not too deep,

Now how do women manage something so complicated? On reflection probably because it is so complicated

In my experience, carrot cake batter is fairly loose and wet, and needs baking for a long time at a reasonably low temperature. I'd imagine that gas mark 4 for 2 hours should be enough for a 9" diameter cake but a deep cake will take longer. Got to say I did have a terrible disaster with carrot cake once, I had too much batter in the tin and it just wouldn't set.

Generally, when a cake is done it will shrink in from the sides of the tin (preparing your tin properly is important!) and if you press it gently, the indentation made by your finger should spring back. A wooden cocktail stick should come out of the cake clean.

Actually, we made carrot muffins in class today and were told to grate them just before mixing up the batter, and they should be grated on the small holes. If you grate them too far in advance then they can actually make the batter too moist - this way the water/juiciness (and flavour) is retained in the grated carrot and doesn't leech out into the batter.

(I'm still reconciling that one in my head, but my instructor knows what he's doing so I'll take his word for it for now).

Also if you're using large eggs, the insides (w/o shell) will weigh 50g - the shells weigh 10g. Basically all recipes call for large eggs except for baking formulas that will do it by weight (e.g. 100g egg whites) or percentage (e.g. 10%, usually relating the ratio of egg to flour). But that's probably more information than you need really :)

Other tips:
- make sure to rotate your pan a few times. All ovens have hot spots and this can help to ensure even cooking
- take the cake out of the tin as soon as possible but don't force it. You don't want the cake to sweat in the tin, but you don't want it to break apart when you take it out either. The more fruit & veg in a baked good (so, carrot cake) the more you'll need to let it settle after baking.