(1) Let $A$ be a commutative ring, and let $M$ and $N$ be $A$-modules. If the natural morphism from $A$ to $\text{End}_A(M\oplus N)$ is surjective, then the annihilators of $M$ and $N$ are comaximal.
Indeed, this comaximality is the condition for the projectors attached the given direct sum decomposition to be in the image.
Assume now that $A$ is a principal ideal domain. Let $(G,+)$ be the Grothendieck group of the category $C$ of finitely generated torsion $A$-modules, and let $(H,\cdot)$ be the group of (nonzero) fractional ideals of $A$.
There is a (clearly unique) morphism from $G$ to $H$ which maps $A/\mathfrak a$ to $\mathfrak a$.
It is easy to see that the fractional ideal attached to $M\in C$ is in fact integral. Call it the characteristic ideal of $M$. Moreover we have in view of (1):
(2) The natural morphism from $A$ to $\text{End}_A(M)$ is surjective, if and only if the characteristic ideal of $M$ coincides with the annihilator of $M$.
Assume now that $A=K[X]$, where $K$ is a field and $X$ an indeterminate, and that $V$ is a finite dimensional $K$-vector space equipped with an endomorphism $a$. Then the characteristic ideal of $V$ is generated by the characteristic polynomial of $a$, and (1) and (2) imply:
The characteristic polynomial of $a$ coincides with its minimal polynomial if and only if any endomorphism of $V$ commuting with $a$ is in $K[a]$.
I don't think the answer to the first question is known.
Will has already pointed out the trivial answer to the second question. However this is not the right question. I mean this is kind of trivial. The interesting question is if you fix the genus and require that the curve over $K$ has good reduction everywhere (outside a fixed set of primes). If you ask it in that way, then the answer for curves is negative (by Faltings) and so the easy fix to do it in higher dimensions does not work.
Here are some comments and references to Theorems 1,2,3:
Theorem 1 is known in more general context.
It does not need "strong", non-isotrivial is enough.
Relevant references are:
Kovács, Sándor J.(1-UT)
Smooth families over rational and elliptic curves.
J. Algebraic Geom. 5 (1996), no. 2, 369–385.
Kovács, Sándor J.(1-MIT)
On the minimal number of singular fibres in a family of surfaces of general type.
J. Reine Angew. Math. 487 (1997), 171–177.
Kovács, Sándor J.(1-CHI)
Algebraic hyperbolicity of fine moduli spaces.
J. Algebraic Geom. 9 (2000), no. 1, 165–174.
Viehweg, Eckart(D-ESSN); Zuo, Kang(PRC-CHHK)
On the isotriviality of families of projective manifolds over curves.
J. Algebraic Geom. 10 (2001), no. 4, 781–799.
Kovács, Sándor J.(1-WA)
Logarithmic vanishing theorems and Arakelov-Parshin boundedness for singular varieties.
Compositio Math. 131 (2002), no. 3, 291–317.
There are also generalizations for families over higher dimensional bases. See for instance:
Viehweg, Eckart(D-ESSN); Zuo, Kang(PRC-CHHK)
Base spaces of non-isotrivial families of smooth minimal models. Complex geometry (Göttingen, 2000), 279–328, Springer, Berlin, 2002.
Kebekus, Stefan(D-KOLN); Kovács, Sándor J.(1-WA)
Families of canonically polarized varieties over surfaces. (English summary)
Invent. Math. 172 (2008), no. 3, 657–682.
Kebekus, Stefan(D-FRBG); Kovács, Sándor J.(1-WA)
The structure of surfaces and threefolds mapping to the moduli stack of canonically polarized varieties.
Duke Math. J. 155 (2010), no. 1, 1–33.
Patakfalvi, Zsolt(1-PRIN)
Viehweg's hyperbolicity conjecture is true over compact bases. (English summary)
Adv. Math. 229 (2012), no. 3, 1640–1642.
Theorem 2:
This is a triviality unless you fix some invariants. On the other hand for relative dimension $1$ and fixed genus this is not true. This is the geometric version of Shavarevich's conjecture and was first proved by Parshin:
Paršin, A. N.
Algebraic curves over function fields. I. (Russian)
Izv. Akad. Nauk SSSR Ser. Mat. 32 1968 1191–1219,
and then in a more general case by Arakelov:
Arakelov, S. Ju.
Families of algebraic curves with fixed degeneracies. (Russian)
Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 1269–1293.
In higher dimensions, the statement is true indeed by taking the product of an arbitrary family of curves and an arbitrary curve (each of the appropriate genus). The second curve can be moved in moduli which gives even a continuous family of families.
In fact, this was what led to the notion of strong isotriviality.
Some relevant references are:
Kovács, Sándor J.(1-WA)
Strong non-isotriviality and rigidity. Recent progress in arithmetic and algebraic geometry, 47–55,
Contemp. Math., 386, Amer. Math. Soc., Providence, RI, 2005.
Kovács, Sándor J.(1-WA)
Subvarieties of moduli stacks of canonically polarized varieties: generalizations of Shafarevich's conjecture. Algebraic geometry—Seattle 2005. Part 2, 685–709,
Proc. Sympos. Pure Math., 80, Part 2, Amer. Math. Soc., Providence, RI, 2009.
Kovács, Sándor J.(1-WA); Lieblich, Max(1-WA)
Boundedness of families of canonically polarized manifolds: a higher dimensional analogue of Shafarevich's conjecture. (English summary)
Ann. of Math. (2) 172 (2010), no. 3, 1719–1748.
Zsolt Patakfalvi
Arakelov-Parshin rigidity of towers of curve fibrations, connections to the infinitesimal Torelli problem
http://arxiv.org/abs/1010.3069
Theorem 3:
as I explained above, even this is not true without the "strong" assumption.
For strongly non-isomorphic families it is proven in
Kovács, Sándor J.(1-WA); Lieblich, Max(1-WA)
Boundedness of families of canonically polarized manifolds: a higher dimensional analogue of Shafarevich's conjecture. (English summary)
Ann. of Math. (2) 172 (2010), no. 3, 1719–1748.
I would expect it to be true for a somewhat larger class of families, but the actual class still needs to be defined. The key modulo this paper is rigidity.
For more details see
Kovács, Sándor J.(1-WA)
Subvarieties of moduli stacks of canonically polarized varieties: generalizations of Shafarevich's conjecture. Algebraic geometry—Seattle 2005. Part 2, 685–709,
Proc. Sympos. Pure Math., 80, Part 2, Amer. Math. Soc., Providence, RI, 2009.
or
Chapter III of
Hacon, Christopher D.(1-UT); Kovács, Sándor J.(1-WA)
Classification of higher dimensional algebraic varieties.
Oberwolfach Seminars, 41. Birkhäuser Verlag, Basel, 2010. x+208 pp. ISBN: 978-3-0346-0289-1
Best Answer
My paper with Bjorn Poonen (which is referenced and discussed in Bjorn's answer to this MO question: Are most cubic plane curves over the rationals elliptic?) has a precise statement for plane curves. You can follow Mike's suggestion in his comment to make a statement for all curves, but this has a problem. Namely, the moduli space of curves of genus $g$ is of general type for $g>22$ (or something like that) so, if you believe Lang's conjecture (or some weakening of it) then there no (or very few) "general" curves of genus $g$ defined over $\mathbb{Q}$, so one expects that most curves of genus $g$ defined over $\mathbb{Q}$ are restricted to rational subvarieties of the moduli space and the biggest one is the hyperelliptic locus, so maybe in some weird sense "most" curves over $\mathbb{Q}$ are hyperelliptic.