This is not really an answer - which I leave to the experts - just a (possibly useless) thought. Let $R=k[x_1,\dots,x_n]$ and $I=(Q_1,\dots,Q_s)\subset R$ the ideal defining $V$. Each $V_i$ corresponds to a prime ideal $\mathfrak p_i\subset R$ (minimal above those) containing $I$. Let us set $\mathfrak p=\mathfrak p_1$ and work with $V_1=V(\mathfrak p)$ where, as we said, $\mathfrak p=(f_1,\dots,f_r)\supset I$.
Now, the first question is about primary decomposition of $I$. The number $m$ is the number of minimal primes in Ass$_R(R/I)$. I don't know whether there is a bound on $m$. Of course it is $\leq |\textrm{Ass}_R(R/I)|$, but I do not feel like this number could depend on the degree of the $Q_j$'s. Indeed (but it's just a naive idea), consider the integers $\mathbb Z$ and look at some $n=\prod_{i=1}^Mp_i^{e_i}$ where the $e_i$'s can be very large powers, and $p_i$ are prime numbers. Well, if $l=\prod_{i=1}^Mp_i$, then $(n)$ and $(l)$ have the same associated primes, and the number $M$ of such associated primes doesn't depend on how huge the generator $n$ is (whence my thought that, in our situation, $m$ should not depend on how huge is the degree of the generators).
For your second question, I can only observe that for every $j$ one has $Q_j\in\mathfrak p$ so $Q_j=\sum_{i=1}^ra_if_i^{b_i}$ so that $\deg Q_j=\max\,(b_i\cdot\deg f_i)\geq\deg f_h$ for all $1\leq h\leq r$.