The key reference for this is MR0799042 (87d:03141). Henle, J. M.; Mathias, A. R. D.; Woodin, W. Hugh. A barren extension. In Methods in mathematical logic (Caracas, 1983), C. A. Di Prisco, editor, 195–207, Lecture Notes in Math., 1130, Springer, Berlin, 1985. There, Henle, Mathias, and Woodin start with $L(\mathbb R)$ under the assumption of determinacy (an […]
This is consistent, at least under a rather tame large cardinal assumption. (One can also produce examples by manipulating Dedekind finite sets, but Asaf's answer addresses this. The answer here works even in the context of $\mathsf{DC}$.) For instance, see MR3612001. Conley, Clinton T.; Miller, Benjamin D. Measure reducibility of countable Borel equiva […]
The only reference I know for precisely these matters is the handbook chapter MR2768702. Koellner, Peter; Woodin, W. Hugh. Large cardinals from determinacy. In Handbook of set theory. Vols. 1, 2, 3, 1951–2119, Springer, Dordrecht, 2010. (Particularly, section 7.) For closely related topics, see also the work of Yong Cheng (and of Cheng and Schindler) on Harr […]
As other answers point out, yes, one needs choice. The popular/natural examples of models of ZF+DC where all sets of reals are measurable are models of determinacy, and Solovay's model. They are related in deep ways, actually, through large cardinals. (Under enough large cardinals, $L({\mathbb R})$ of $V$ is a model of determinacy and (something stronge […]
Forcing with sufficiently homogeneous forcing that adds reals is enough to obtain the negation of $(*)$. The point is that if a formula $\phi$ defines a parameter-free well-ordering of $\mathbb R$, then for any ordinal $\alpha$, the statement "$x$ is the $\alpha$-th real in the well-ordering defined by $\phi$" uniquely characterizes $x$ in terms of […]
This is an excellent question! Note first that, just from cardinal arithmetic considerations, whether we obtain all sets is independent (and therefore so is whether we obtain all Lebesgue measurable sets). The right context to study this question is descriptive set theory, and indeed the problem was considered early on. For example, Sierpiński proved in Sur […]
Surprisingly, the answer is no due to serious set-theoretic restrictions. If $X$ is an infinite set such that there is a ($\sigma$-additive) measure on $\mathcal P(X)$ that only takes the values 0 and 1, assigns 1 to $X$ and 0 to singletons, then the cardinality $\kappa=|X|$ of $X$ is much much larger than $\mathfrak c=|\mathbb R|$, the cardinality of the se […]
No, you cannot show this. For instance, it is consistent to have infinite Dedekind-finite sets whose power set is still Dedekind-finite. Now, if there is a surjection from $A$ to $\omega$, then there is an injection from $\omega$ (indeed, from $\mathcal P(\omega)$) to $\mathcal P(A)$, so $\mathcal P(A)$ is Dedekind-infinite. Thus, if $\mathcal P(X)$ is infin […]
First, there are some nice examples like $$ e=\sum_{n\ge0}\frac1{n!} $$ or Liouville-like numbers, mentioned in the answers by Wilem2, that can be easily proved to be irrational using the theorem, but for which typically there are simpler irrationality proofs: For $e$, we quickly get that $$0