(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.
I will offer some words on this, but only because no-one else has; I was holding out hoping that one of the more automorphic people would chip in. It might be worth taking much of the below with a pinch of salt.
So I've been trying to penetrate the "fundamental lemma" literature myself, and let me begin by showing my hand and saying that my current impression, that may be wrong, is that "the fundamental lemma" is not a well-defined mathematical statement, it is a principle that applies in many situations, and I think that mathematics is open to the situation where there will be precise mathematical statements formulated in the future, when people are studying situation $X$, and that people will call these statements "the fundamental lemma for $X$". For example, I think that perhaps in 50 years' time when people are proving general base change (rather than cyclic base change) there may be a "general base change fundamental lemma".
So this of course immediately raises the question -- what did Ngo Bao Chau do? Well, my perception is that he has proved the version of the fundamental lemma that shows up in the theory of endoscopy.
As I've already tried to make clear, I am not 100% sure on all of this. But let me muddle on regardless. My impression is that what the game looks like is this. There is the general functoriality principle, which says that if I have two connected reductive groups $G$ and $H$ over a global field, and a map between the $L$-group ${}^LH\to{}^LG$, then there should be some sort of way of transferring automorphic representations on $H$ to automorphic representations on $G$. Already this is a "principle" rather than a precisely-formulated statement, because I think there are lots of issues with packets and multiplicities, that certainly I don't understand, when trying to make a precise statement, and I am not sure I've seen a precise statement in the literature that ties up all the loose ends that I want to see tied up, other than some very weak ones that just demand compatibility at the unramified places and don't care about multiplicities at all.
But it turns out, when attempting to e.g. study the zeta functions of some Shimura varieties, that Langlands needed some very special cases of the functoriality principle, where $H$ was an endoscopic subgroup for $G$, and here he postulated a strategy for proving functoriality using the trace formula. The idea is that you try and stabilise the trace formula for $G$, whatever that means, and this involves, amongst other things, figuring out some way of transferring functions locally and matching the orbital integrals that come up. The upshot is that you relate the trace formula for $G$ to the trace formula for all the endoscopic subgroups for $G$.
So my perception is that, in its initial state, the situation was this: $G$ was a connected reductive group, now over a local field, $H$ an endoscopic subgroup (and part of the data of this endoscopic situation is that you're given a map ${}^LH\to{}^LG$) and there is supposed to be a "transfer" map, that maps functions on $G$ to functions on $H$. Even the existence of the transfer map is not at all clear in general, I don't think. For example Labesse-Langlands have to do some calculations (only a few pages, but some work) to prove that one can transfer functions from $SL(2)$ to a subtorus (this is the simplest example of endoscopy, I think).
So my impression is that the general notion of moving from functions on one group, locally, to functions on another, is called "transfer of functions". My understanding is that the transfer map is not at all well-defined, that sometimes one can characterise the image (as being functions whose orbital integrals vanish on some certain subgroup), and that at the end of the day the precise relationship you want between the function and its transfer can be quite complicated. I think that in general you want the orbital integral of one function to be the orbital integral of the other multiplied by a "fudge factor" whose definition is the key point of a 100-page paper by Langlands and Shelstad. One can already see these fudge factors in the $SL(2)$ case with Langlands-Labesse.
My understanding of what the fundamental lemma is, is the following: in the situation where $G$ and $H$ are unramified, one extra condition you could put on the "transfer of functions" map is that the identity element for the unramified Hecke algebra for $G$ gets sent to the identity element for the unramified Hecke algebra for $H$. Hence in this situation, "the fundamental Lemma", I think, boils down to the assertion that a certain volume equals a fudge factor that it takes 100 pages to define, multiplied by another volume.
I'm slowly getting to the point :-) I think I can answer one of your questions at least -- the "buzzword" you're looking for is not "fundamental lemma" but "transfer of functions" or perhaps "local transfer of functions". I think.
However, my understanding is that the situation you are looking at is not an "endoscopic situation". In particular I don't think that reading the complete works of Ngo Bao Chau will give you an answer to your question. You have groups $G$ and $G'$ and they're both $GL(n)$ but over different fields, so if I were trying to prove a hard global theorem and I needed the type of transfer that you're looking for, I would probably not be trying to stabilise the trace formula (indeed I think the trace formula for $GL(n)$ is already stable and that "$GL(n)$ has no endoscopy" in some sense) -- I would probably be trying to prove base change.
Now here are, for me, some BIG problems I would fear when attempting to try and get an answer to your question from the literature that I know about.
The first is that, when trying to prove global base change for $GL(n)$ for a global cyclic extension $L/K$, I attempt to find a relation between the trace formulas for $GL(n)$ over $L$ and over $K$ and then I attempt to start matching up terms etc etc, and the problem is that one trace formula is a sum over conj classes in $GL(n,L)$ and the other is a sum over $GL(n,K)$. As far as I know, people don't know how to relate these two sets in a natural way. So what they do is they relate conjugacy classes in $GL(n,K)$ to twisted conjugacy classes in $GL(n,L)$. I think the local story looks like this: say $E/F$ is now local and has cyclic Galois group. Given $\gamma$ in $GL(n,E)$ they take its "norm" in the most naive way (multiply it by its Galois conjugates) and get an element of $GL(n,F)$ and in this nice cyclic situation they can inject twisted conj classes ($x\sim \sigma(g)xg^{-1}$ with $\sigma$ a generator of the Galois group) in $GL(n,E)$ with conj classes in $GL(n,F)$. Using this trick they want to relate the usual trace formula for $GL(n,K)$ with a twisted trace formula for $GL(n,L)$, and to do this the transfer of functions they require is a twisted transfer! In particular, they need a machine which, given a function $a$ on $GL(n,E)$ spits out a function $b$ on $GL(n,F)$ such that the orbital integrals of $b$ equal certain twisted orbital integrals for $a$, possibly again multiplied by some fudge factors, but I believe that in this base change situation the fudge factors (which are I think not covered by the Langlands-Shelstad monster because this is not an endoscopic situation) are all 1 anyway.
The upshot is that in the Arthur-Clozel book, proving cyclic base change for $GL(n)$, you will see a definition of transfer of functions, which looks formally a bit similar to the thing you write above, but there are certain crucial differences:
1) if $a$ transfers to $b$ then an orbital integral for $b$ will equal a twisted orbital integral for $a$, rather than an orbital integral.
2) The twisted orbital integral for $a$ will be attached to an element $\gamma$ and the orbital integral for $b$ that it equals will be attached not to $\gamma$ but to its naive norm (which is a well-defined conj class, I believe, in $GL(n,F)$)
3) $E/F$ will ALWAYS be cyclic (or perhaps more generallu they will allow $E=F^n$ so $E$ is not actually a field but it's still etale over $F$).
Now I look at the question you're asking, and there are of course two ways of approaching it: the first is to try and get your hands dirty and write down the map between functions yourself. But it sounds to me like you're hoping that you can take another approach -- to get what you want from the literature. And what I am really scared by is that although what you write looks to me superficially like transfer of functions, you are in a situation where $F'$ is not in general a cyclic Galois extension of $F$, so you can throw away Arthur-Clozel, and you are not I think in an endoscopic situation either, so you can also throw away Ngo Bao Chau, and unfortunately I personally do not know what is left. Of course this will largely be my own ignorance and probably there are people who have thought about "transferring" in its own right, independent of links to functoriality. But I am now not sure where to point you.
Aah, the joyful tones of my daughter, who has apparently just woken up. What timing she has! I've gotta go, but I've said all I can say anyway.
Best Answer
Now I think the answer is yes, Arthur's work is now unconditional for quasi-split special orthogonal and symplectic groups. Wee Teck Gan kindly directed me to the relevant papers of Waldspurger and Moeglin addressing the assumptions 2-4 above, though I was not able to verify with certainty that their stated results precisely cover what Arthur requires. Here are the relevant papers:
Edit: Arthur confirmed that the results in his book (except the last chapter on non-quasi-split forms) are now unconditional.