Yes, it holds also for $p=\infty$ and, more general, when $L^p$ is replaced by an ideal space and when $\mathbb R_{\ge0}$ is replaced by a Banach space. For instance, it also holds for every Orlicz space (which in general is non-separable if the generating Young function fails to satisfy the $\Delta_2$ condition). Also $\mathbb R^d$ can be replaced by any $\sigma$-finite measure space. A proof is given in the reference from the link. Note, however, that "measurable" needs to be defined as "Bochner measurable" (which implies essentially ("a.e.") separable range).

Yes, see mathoverflow.net/questions/67434/…

@TimothyChow: In nonstandard analysis of Robinson/Luxemburg, the transfer principle applies, roughly speaking, only to standard sentences; but there are more. You can “explicitly” define non-measurable sets there. For instance, let $f\colon[0,1)\times\mathbb N\to\{0,1\}$ be such that $f(x,n)$ is the $n$-th digit of the binary expansion of $x$. Then for any infinite hypernatural $h$ the set $\{x\in[0,1):{}^*f({}^*x,h)={}^*1\}$ is not Lebesgue measurable. It is surprising that this set cannot be defined in the restricted languages of internal set theory from the cited paper of Hrbacek-Katz.

@TimothyChow ”almost impossible” was probably a bit too harsh, but you have to be extremely careful which definitions you choose. So there is not “the” integration theory anymore in ZF, but several such theories, usually preserving different properties (compared to the ZF+DC situation which has all properties and in which all approaches are equivalent, at least for the Lebesgue measure).

It depends how one interprets it: If you consider ZF+DC as the natural basis, then removing AC from (ZF+DC)+AC (where DC is as redundant as some other axioms are in many formulations of ZF), you end up with ZF+DC. (I claim that ZF+DC is a more natural basis for mathematics than ZF alone, but opinions may vary.)

@SergeiAkbarov Note that DC is not a replacement of AC; it is just a weak form of AC which suffices to exclude many counter-intuitive (from a practical point of view) examples which are not non-provable in ZF and is simultaneously not powerful enough to prove some counter-intuitive theorems which are provable in ZF+AC (like Hausdorff-Banach-Tarski). As an ”alternative” to AC, I would instead consider something like ZF+DC+“all subsets of R have the Baire property”,

@SergeiAkbarov “people usually don't discuss the possibility to replace AC, they are speaking about its removing” - yes, but what ”removing” means depends on what one considers as a base theory. This will be ZF for a “pure” set theorist, but ZF+DC for the “practical man”. Unfortunately, after the groundbreaking work of Solovay and Pincus in the 70s, it has become less popular under set theorist to investigate what cannot be proved if ZF+DC.

@SergeiAkbarov: The question of whether something holds without AC usually means whether this something holds in every universe satisfying the same axioms except AC. By Gödel's completeness theorem, it is “usually” (e.g. when we speak about ZF or ZF+DC as the underlying axioms) equivalent to ask whether it can be proved without AC (of course, including all lemmas necessary for the proof). You are mistaken if you think that almost nothing can be shown without AC. At least in ZF+DC most statements from analysis can be proved except for some “exotic” ones.

Divide $[0,1]$ into $k$ disjoint sets of measure less than $\varepsilon$ and choose $k$ times iteratively subsequences for each of these sets.

The proof is not from me. IIRC, Luxemburg had observed it already in the 60's in some paper in which he proved the Tychonoff theorem for Hausdorff spaces using only the boolean prime ideal theorem instead of the axiom of choice: For Hausdorff spaces, st is a function, and so no additional invoking of AC is necessary - having a nonstandard model is sufficient, and for that in turn the boolean prime ideal is sufficient, at least in this case.

No, I do not know a standard reference. Usually a good general reference in that area is the Springer lecture notes of Castaing & Valadier, but it may be too old for that (and is very hard to read, also because of the typesetting). As mentioned, perhaps the monograph of Diestel & Uhl might contain the result. By "the earlier mentioned theorem" I meant the theorem about weak compactness (what you called Diestel theorem) which you are looking a reference for. But if you are really only interested in the $\mathbb R^n$ case, I would check whether the proof in the Aumann paper is not enough.