"J'ai besoin de lait" Is 'de lait' actually the contraction of 'de + du'? (de+du=de)

No, it is actually the other way around: du = de + le. There is no contraction of "de+du"; that wouldn't make any sense.

The phrase here is: avoir besoin de quelque chose. Therefore you use de.

I think Chris and Cécile may have missed the point of the question.

"I need" = "j'ai besoin de"

"some milk" = "du lait".

But you don't say "j'ai besoin de du lait", so, in a sense, "de du" contracts to "de".

Alternatively, if you wanted to say "I need the milk" - pointing to some specific milk

"the milk" = "le lait"

You don't say "j'ai besoin de le lait", but "j'ai besoin du lait", so here "de le" contracts to "du".

I agree with both Joan's and Alan's logic with regard to the contractions after preposition "de"

The plural indefinite article and the partitif are ommited as follows

de + des, de+du, de+de la, de + de l' all become de

e.g. Ce verre est pleine d'eau (-de +de l'eau)

J'ai besoin de clous (de + des clous)

Faute de temps, j'ai raté le match (de + du temps)

Referring to logic can aid in choosing the form and avoiding substituting erroneous indefinite and partitif articles.

Actually Alan's answer cleared up something that has been bugging me.

Why "l'air du temps" and not "l'air de temps" obviously we are referring ta specific time, ie NOW, the present.