I bit long for a comment, so posted as an answer, although this is really a comment.

For 1) I would rather try the following. Assuming the $f$'s have the same degree in $x_1$, introduce formal variables $s_1,\ldots,s_n$ and $t_1,\ldots,t_n$, and take the binary resultant ${\rm Res}(s_1f_1+\cdots,s_nf_n,t_1f_1+\cdots+t_nf_n,x_1)$. This is a monstruous polynomial in the $s,t$ variables. I think the coefficients of this polynomial should be in $I_1$ and may give a basis. If the $f$'s don't have the same degree, you may have to make two copies of them, multiply one copy by powers of $x_1$ and the other copy by powers of $1-x_1$, so they all have the same degree, then do the above with $2n$ instead of $n$.

A bit long for a comment, so posted as an answer, although this is really a comment.

For 1) I would rather try the following. Assuming the $f$'s have the same degree in $x_1$, introduce formal variables $s_1,\dotsc,s_n$ and $t_1,\dotsc,t_n$, and take the binary resultant $\operatorname{Res}(s_1f_1+\dotsb+s_nf_n,t_1f_1+\dotsb+t_nf_n,x_1)$. This is a monstrous polynomial in the $s,t$ variables. I think the coefficients of this polynomial should be in $I_1$ and may give a basis. If the $f$'s don't have the same degree, you may have to make two copies of them, multiply one copy by powers of $x_1$ and the other copy by powers of $1-x_1$, so they all have the same degree, then do the above with $2n$ instead of $n$.