(1) Regarding the relationship between geometric Langlands and function field Langlands:
typically research in geometric Langlands takes place in the context of rather restricted ramification (everywhere unramified, or perhaps Iwahori level structure at a finite number of points). There are investigations in some circumstances involving wild ramification (which is roughly the same thing as higher than Iwahori level), but I believe that there is not a definitive program in this direction at this stage.
Also, Lafforgue's result was about constructing Galois reps. attached to automorphic forms. Given this, the other direction (from Galois reps. to automorphic forms), follows immediatly, via
converse theorems, the theory of local constants, and Grothendieck's theory of $L$-functions in the function field setting.
On the other hand, much work in the geometric Langlands setting is about going from local systems (the geometric incarnation of an everywhere unramified Galois rep.) to automorphic sheaves (the geometric incarnation of an automorphic Hecke eigenform) --- e.g. the work of Gaitsgory, Mirkovic, and Vilonen in the $GL_n$ setting does this. I don't know how much is
known in the geometric setting about going backwards, from automorphic sheaves to local systems.
(2) Regarding the status of function field Langlands in general: it is important, and open, other than in the $GL_n$ case of Lafforgue, and various other special cases. (As in the number field setting, there are many special cases known, but these are far from the general problem of functoriality. Langlands writes in the notes on his collected works that "I do not believe that much has yet been done beyond the group $GL(n)$''.) Langlands has initiated a program called ``Beyond endoscopy'' to approach the general question of functoriality. In the number field case, it seems to rely on unknown (and seemingly out of reach) problems of analytic number theory, but in the function field case there is some chance to approach these questions geometrically instead. This is a subject of ongoing research.
This is a "big-picture" question, but allow me to illustrate some recent progress by taking a small example close to my heart.
Let us adjoin to the field $\mathbb{Q}_p$ a primitive $l$-th root of $1$, where $p$ and $l$ are primes, to get the extension $K|\mathbb{Q}_p$. We notice that this extension is unramified if $l\neq p$ but ramified if $l=p$. When we adjoin all the $l$-power roots of $1$, we get the $l$-adic cyclotomic character $\chi_l:\operatorname{Gal}(\bar{\mathbb{Q}}_p|\mathbb{Q}_p)\to\mathbb{Q}_l^\times$ which is unramified if $l\neq p$ but ramified if $l=p$. But we cannot just say that $\chi_p$ is ramified and be done with it. We have to somehow express the fact that $\chi_p$ is a natural and a "nice" character, not an arbitrary character $\operatorname{Gal}(\bar{\mathbb{Q}}_p|\mathbb{Q}_p)\to\mathbb{Q}_p^\times$, of which there are very many because the topologies on the groups $\operatorname{Gal}(\bar{\mathbb{Q}}_p|\mathbb{Q}_p)$, $\mathbb{Q}_p^\times$ are somehow "compatible".
The fact that $\chi_p$ is a "nice" character is expressed by saying that it is crystalline. In general, we can talk of crystalline representions of $\operatorname{Gal}(\bar{\mathbb{Q}}_p|\mathbb{Q}_p)$ on finite-dimensional spaces over $\mathbb{Q}_p$; the actual definition is in terms of a certain ring $\mathbf{B}_{\text{cris}}$, constructed by Fontaine, which can be understood in terms of crystalline cohomology.
My illustrative example is about the $l$-adic criterion for an abelian variety $A$ over $\mathbb{Q}_p$ to have good reduction. For $l\neq p$, this can be found in a paper by Serre and Tate in the Annals, and it is called the Néron-Ogg-Shafarevich criterion. It says that $A$ has good reduction if and only if the representation of $\operatorname{Gal}(\bar{\mathbb{Q}}_p|\mathbb{Q}_p)$ on the $l$-adic Tate module $V_l(A)$ is unramified.
What happens when $l=p$ ? It is too much to expect that $V_p(A)$ be an unramified representation when $A$ has good reduction; we have seen that even $\chi_p$ is not unramified. What Fontaine proved is that the $p$-adic representation $V_p(A)$ is crystalline (if $A$ has good reduction). To complete the analogy with the case $l\neq p$, Coleman and Iovita proved in a paper in Duke that, conversely, if the representation $V_p(A)$ is crystalline, then the abelian variety $A$ has good reduction.
I hope you find this enticing.
Best Answer
The two formulas in the abstract were proven by relatively simple methods in a couple of days after the paper appeared on arxiv. See https://arxiv.org/abs/1907.05563 The rest of the formulas inside the paper have not been proven so far, to the best of my knowledge.