Learn English – Singular or Plural nouns

The correct use changes depending on the sentence:

Do you have any idea how to do this?
Do you have any idea what to do?
Do you have any ideas for me?
Do you have any ideas for how to do this?

It seems that, if the singular or plural noun (idea) is directly connected by a subordinating conjuction (how / what / where / which / that), you use the singular; if it's modified by a prepositional phrase (for...) or by nothing at all, you use the plural.

Without using all that grammar jargon, I'd say use the singular if it's followed by how, what, that, who, whom, whose, which, etc, and use the plural otherwise.

This is a very good question. In general, you can help yourself work through any confusion by trying to express the idea more precisely or in a slightly different way. I'll use each of your examples to explain this a bit further.

EX1: All lizards have a head width, but that isn't what you measured. You measured the different head widths of a group of lizards, or "the head widths of some lizards." I would at least add the "some", but perhaps it would be better to include the actual number.

EX2: You are talking about the growth rate of all birds, not, say, the growth rates of different species of birds.

EX3: Each of two populations has a mean height, therefore there are two mean heights. You are saying that the mean heights of these two populations are different.

EX4: Dogs have noses is correct, but this is a bit inconsistent. For example, dogs have a sixth sense that lets them know when someone is coming, and they have a nose for trouble. People have minds, but they have a combative streak. So, when you are talking about a shared characteristic of all dogs, you use the singular, but when you are talking about something that each dog has, you use the plural.