How well do subspaces hit sets
Here's an $\Omega(\log(\frac{1}{\epsilon\cdot \delta}))$ lower bound following the discussion here with Yuval, and another one with Ross Berkowitz.
I assume for simplicity that the sampling process chooses $k$ completely random elements $v_1,\ldots,v_k$ of $F_2^n$ and outputs their span $V$.
A little extra work is needed to bound the probability that the $k$ vectors sampled are not independent, and in fact, this might be problematic when $k$ is close to $n$.
Denote $\ell\triangleq \log(1/\epsilon)$, and let $S$ be an affine subspace of co-dimension $\ell$. Let $S_0$ be the linear subspace of which $S$ is a shift.
Denote by $T\triangleq F_2^n/S_0$ the quotient space, which is a linear space of dimension $\ell$ who's elements are the cosets of $S_0$ in $F_2^n$. For each $v_i$ denote by $\bar{v}_i\in T$, the coset in which $v_i$ landed.
Denote by $T'\subset T$ the linear span of $\bar{v}_1,\ldots,\bar{v}_k$.
Assume $k\triangleq (\ell-1) + t$, we can lower bound the probability of the event $T' \subsetneq T$ by $2^{-t}$:
Condition on $\bar{v}_1,\ldots,\bar{v}_{\ell-1}$ being linearly independent - this can only decrease the probability of $T'\subsetneq T$.
Denote $T''\triangleq \mathrm{span}(\bar{v}_1,\ldots,\bar{v}_{\ell-1})$
Now, there is a probability of at least $2^{-t}$ that the rest of the $\bar{v}$'s all land in $T''$, and in such a case $T'\subsetneq T$
Now, note that $V\cap S \neq \emptyset$ is exactly equivalent to $x\in T'$ for some fixed $x\in \{0,1\}^{\ell}$ (that correspond to the what coset $S$ is of $S_0$). Furthermore, conditioned on the event $T'\subsetneq T$ $x\notin T'$ with probability at least $1/2$. Thus $V\cap S = \emptyset$ with probability at least $2^{-t}$. So we need $2^{-t}\leq \delta$ which implies $k\geq (\ell-1) + \log(1/\delta) = \log(1/\epsilon) + \log(1/\delta) -1$.