It is easy to show that a complex polynomial with $N$ non-zero coefficients cannot have a non-zero root of multiplicity $N$ or more. Is there any standard name / reference for this fact?
Also, observing that $(x+a)^{N-1}$ has $N$ non-zero coefficients and a root of multiplicity $N-1$, can the assertion above be strengthened given that the polynomial in question is not of this form?
The small print: posting on behalf of a friend; also posted at MathStackExchange about a week ago.
I'll start by answering the question about polynomials with $N$ non-zero coefficients and a non-zero root of multiplicity $N-1$. Polynomials of the form $(x-a)^{N-1}$ are not the only such polynomials!
Theorem: Let $m_{N}>m_{N-1}>\cdots >m_1\geq 0$ be $N$ given integers, and $a$ some non-zero complex number. There exists a unique monic polynomial $$f(x)=\sum_{i=1}^{N} c_ix^{m_i}$$ which has $a$ as a root of multiplicity $N-1$.
Proof: The coefficients $c_i$ must satisfy $c_{N}=1$ and $N-1$ linear equations from $\frac{d^i}{dx^i}f (a)=0$ for $i=0,1,\dots,N-2$. If we define the matrix $$A=\Big((j-1)!\binom{m_{N-i}}{j-1}a^{m_{N-i}}\Big)_{i,j=1}^{N-1}$$ our equations in the variables $v=(c_{N-1},\dots,c_1)$ can be writen as $$Av^{T}=w$$ where $w=-a^{m_N}(1,m_N,2\binom{m_N}{2},\dots,(N-2)!\binom{m_N}{N-2})$. Finally notice that $A$ is invertible since it can easily be modified to a Vandermonde type matrix (like here).
Now, as far as your first question I have never seen these type of lemmas be called by a specific name, and they seem to be folklore. A recent article that talks about various generalizations is "Root multiplicities and number of nonzero coefficients of a polynomial" which shows that these type of results have analogues in positive characteristic. You can find a couple of other references in there, as well.
