Suppose $A:=\{f_1,\dots,f_m\}\subset \mathbb{C}[x_1,\dots,x_n]$ with $m>n$ is a set of homogeneous polynomials of equal degree $d>0$. Suppose further that the variety they define consists of a single point (i.e. the zero vector).

Is it true that there exists a subset $g_1,\dots,g_n$ of the $\mathbb{C}$-linear span of $f_1,\dots,f_m$ with the same property? I.e. they have exactly one common zero?

That this is true for $d=1$ is basic linear algebra. For general $d$, my geometric intution (which, I admit, is probably rather bad given my inexperience in algebraic geometry), tells me that any subset $g_1,\dots,g_n$ "in general position" should have the desired property, which is why I believe this statement to be true.

Thus, two "bonus questions":

Is it true that any "sufficiently generic" subset $g_1,\dots,g_n$ does it, and that "almost all" subsets of size $n$ are "sufficiently generic" in some suitable sense?

Finally (this holds for $d=1$, I think):

Can one always choose $g_1,\dots,g_n\in A$?

I wouldn't be surprised if for somebody who is familiar with algebraic geometry, this is a basic exercise, and so I apologize if this is deemed unsuitable for MO. I'd still be grateful for any references, proofs or hints.