For question 1, the following example works, and you'll see how to construct many more (in higher characteristic).

Set $k$ to be a perfect field of characteristic $2$.

$$X = \text{Proj } k[x,y,u,v]/\langle x^3+y^3+u^3+v^3 \rangle \subseteq \mathbb{P}^3_k.$$

### Fano check:

I'm certain you already know this but... Certainly $K_X \sim (K_{\mathbb{P}^3} + X)|_X \sim (-4H + 3H)|_X = -H|_X$ where $H$ is the hyperplane on $\mathbb{P}^3_k$. Clearly also $X$ is smooth. Thus $X$ is Fano.

### Frobenius splitting check:

This is slightly more involved. First we need a couple lemmas:

**Lemma:** [Probably due to Karen Smith] *A projective variety $X$ is Frobenius split if and only if some/every section ring with respect to an ample divisor*
$$S_X = \bigoplus_{n \geq 0} O_X(nA).$$
*is Frobenius split.*

The proof is pretty easy. If $X$ is Frobenius split, so is $S_X$ (use the splitting on $S_X$ induced by that on $X$ on degrees divisible by $p$, throw the other degrees out). For the converse direction, it's not hard to show that if $S_X$ has a Frobenius splitting, it then has an appropriately graded Frobenius splitting (basically, use the fact that $Hom_{S_X}(F_* S_X, S_X)$ is generated by appropriately graded maps). It then induces maps on the associated sheaves. Care must be taken since $\widetilde{F_* S_X} \neq F_* \widetilde S_X = F_* O_X$, but the latter is a summand of the former and this is enough.

Now for the next lemma. If $J = \langle g_1, \dots, g_t \rangle$ is any ideal in a ring of characteristic $p$, we use $J^{[p]}$ to denote the ideal $\langle g_1^p, \dots, g_t^p \rangle$.

**Lemma:** [Fedder's Criterion] *Suppose $p = \text{char} k$, then a ring $k[x_1, \dots, x_n]/I$ is Frobenius split at a point $\mathbb{m} \in \text{Spec} k[x_1, dots, x_n]$ if and only if $I^{[p]} : I \nsubseteq \mathbb{m}^{[p]}$.*

See Lemma 1.6 in *$F$-purity and rational singularity* by Richard Fedder (1983, Transactions of the AMS). It's quite easy, but maybe more than I want to explain.

**Corollary:** *If $p = 2$ and $I = \langle f \rangle$, then $R$ is Frobenius split at $\mathbb{m}$ if and only if* $f \notin \mathbb{m}^{[2]}$. *(in other characteristics, you have( $f^{p-1} \notin \mathbb{m}^{[p]}$)*.

Ok, now we verify that $X$ is not Frobenius split. Consider the section ring given our embedding, $S_X = k[x,y,u,v]/\langle x^3 + y^3 + u^3 + v^3 \rangle$. It is sufficient to show that $S_X$ is not Frobenius split, and that we can check at the origin since $S_X$ is a graded ring. Then we simply observe that
$$
x^3+y^3+u^3+v^3 \in \langle x^2, y^2, u^2, v^2 \rangle.
$$

## Generalizations, question 2, and further references

The result you mention about Fanos is actually generalized to log-Fano varieties in this paper: Globally $F$-regular and log Fano varieties due to Karen Smith and myself. Log Fano's are varieties where $-K_X$ is not necessarily ample, but where it is *close*. Some additional discussion of the lemmas above can also be found here.

Lots of people in commutative algebra have been exploring question of when various rings are $F$-split. For example, see THIS PAPER of Daniel Hernandez and references. I know there are people exploring exactly when graded hypersurfaces are $F$-split = $F$-pure but this is not public yet.

In terms of Question 2: effective bounds, I don't know any great ones off the top of my head. I'll try to get back to you on this.