Least number of non-zero coefficients to describe a degree n polynomial

The modern notion of the essential dimension of a group gives a precise way to state your question (and generalizations), and there is some recent work extending the work mentioned in Scott's answer. To get started, see the article

J. Buhler and Z. Reichstein, On the essential dimension of a group, Compositio Math. 106 (1997), 159-179.

For instance, it is proved there that for polynomials of degree $n$, at least $\lfloor n/2 \rfloor$ coefficients are required. (This agrees with what you mentioned for $n=5$ and $n=6$.)

You might have a look at Polynomial Transformations of Tschirnhaus, Bring and Jerrard (Internet Archive). It gives more explicit detail on why you can remove the first three terms after the leading term (covering the cases of degree 5 and 6 you mention above), but it does concentrate on degree 5.

Hamilton's 1836 paper (Internet Archive) on Jerrard's original work has an elementary explanation of the technique (much of the paper concentrates on showing that certain other reductions Jerrard proposed, including a general degree 6 polynomial to a degree 5, were "illusory"). It also explains Jerrard's trick for eliminating the 2nd, 3rd and 5th terms. Finally, Jerrard has a method for eliminating the second and fourth terms, while bringing the third and fifth coefficients into any specified ratio: this only works in degree 7 or above (Jerrard had mistakenly thought this worked generally, and thus solved the general quintic by reducing it to de Moivre's solvable form -- this all predates Abel's work!)

If by "Bring-Jerrard" form you just mean a certain number of the initial terms (after the first) have been eliminated, then the Hamilton numbers you linked to are indeed exactly what you want.