(1) If $s,u$ and $w$ are all multiples of a single vector $a$, then we can write $s=c_1a,u=c_2a$ and $w=c_3a$ for scalars $c_1,c_2$ and $c_3$. Since $t,v$ and $x$ are linearly dependent, we can write one of these vectors as a linear combination of the other two. (Exercise.) Let us assume, without loss of generality, that we can write $x=e_1t+e_2v$ for scalars $e_1$ and $e_2$. The bilinearity of $f$ implies that:
$f(w,x)=f(c_3a,x)=c_3f(a,x)=c_3f(a,e_1t+e_2v)=c_3\left[e_1f(a,t)+e_2f(a,v)\right]$.
Therefore, we know the value of $f(w,x)$ if we know the values of $f(a,t)$ and $f(a,v)$. Exercise: use the bilinearity of $f$ to show that (for $c_1,c_2\neq 0$) we know the values of $f(a,t)$ and $f(a,v)$ if we know the values of $f(s,t)$ and $f(u,v)$.
(2) I think he means that there does not exist a vector $a$ such that $s=c_1a, u=c_2a$ and $w=c_3a$ for scalars $c_1,c_2$ and $c_3$ and similarly there does not exist a vector $b$ such that $t=d_1b,v=d_2b$ and $x=d_3b$ for scalars $d_1,d_2$ and $d_3$.
The following exercises are relevant (two are already stated above but I will state them again (in a more general situation) for the convenience of location):
Exercise 1: Let $(v_1,\dots,v_n)$ be a linearly dependent tuple of vectors in a vector space $V$ for some positive integer $n$. If $v_1\neq 0$, prove that for some $1\leq i\leq n$, we can write $v_i$ as a linear combination of the tuple $(v_1,\dots,v_{i-1})$.
Exercise 2: Let $f:V\times V\to W$ be a bilinear map where $V$ and $W$ are vector spaces. Prove the following:
(a) $f(v_1,0)=0=f(0,v_2)$ for all vectors $v_1,v_2\in V$.
(b) Let $c\neq 0$. If we know the value of $f(cv_1,v_2)$ for vectors $v_1,v_2\in V$ and some scalar $c$, then we know the value of $f(v_1,v_2)$.
Exercise 3: Let $f:V\times V\to W$ be a bilinear map where $V$ and $W$ are vector spaces and let $(v_1,\dots,v_n)$ be a basis of $V$. Prove that if we know the values of $f(v_i,v_j)$ for all $1\leq i,j\leq n$, then we know the value of $f(a,b)$ for any pair of vectors $a,b\in V$.
Exercise 4: Let us maintain the notation of Exercise 3 and let $A$ be the $n\times n$ matrix with $f(v_i,v_j)$ as its $(i,j)$th entry. If $a,b\in V$, prove that $f(a,b)=a^{T}Ab$ where $a^{T}$ is the transpose of the column vector $a$ and $b$ is viewed as a column vector.
Exercise 5: Let $V$ be a vector space of dimension $n$ and let $a_1,\dots,a_n,a_{n+1},b_1,\dots,b_n,b_{n+1}$ be vectors in $V$. If all the $a_i$'s are multiples of a fixed vector, show that at least one of the values $f(a_i,b_i)$ for $1\leq i\leq n+1$ is redundant, i.e., can be deduced with a knowledge of the other $n$ values (plus possibly an understanding of basic properties of bilinear forms such as (a) of Exercise 2).
Challenge: Formulate conditions under which we can deduce the value of a bilinear map at a given ordered pair of vectors if we know the values of the bilinear map on a specific set of ordered pairs. (In other words, generalize Exercise 5 and think more about Exercise 3 and Exercise 4.)
I hope this helps!