The answer to a sharper question involving integers, rather than rationals, is affirmative.
Let $\lambda$ be a positive real algebraic integer that is greater in absolute value than all its Galois conjugates ("Perron number" or "PF number"). Then $\lambda$ is the Perron–Frobenius eigenvalue of a positive integer matrix.
(The converse statement is an integer version of the Perron–Frobenius theorem, and is easy to prove.)
In a slightly weaker form (aperiodic non-negative matrix), this is theorem of Douglas Lind, from
The entropies of topological Markov shifts and a related class of algebraic integers.
Ergodic Theory Dynam. Systems 4 (1984), no. 2, 283--300 (MR)
I don't have a good reference for the strong form, but it was discussed at Thurston seminar in 2008-2009. One interesting thing to note is that, while the proof can be made constructive, it is non-uniform: the size of the matrix can be arbitrarily large compared to the degree of $\lambda$.
I've gained some new perspective on this question, based partly on comments and on Hitachi Peach's answer. Instead of editing the original question, I'll write down some more thoughts as a (partial) answer in the hopes that it will inspire someone with different expertise to say more.
First, after Hitachi Peach's comment following his answer, I tried plotting a picture
of all the answers for a couple two of the simplest situations: quadratics and cubics
with a small value of $C$.
Below is a diagram in the coefficient space for quadratic polynomials. The horizontal axis is the coefficient of $x$ and the vertical axis is the constant.
alt text http://dl.dropbox.com/u/5390048/QuadraticSmallPerron.jpg
The unshaded area in the middle are polynomials whose roots are real with maximum absolute
value 5 and minimum absolute value 1; the left half of this area consists of Perron polynomials. The red lines are level curves of the maximum root.
Here is a similar plot for cubic polynomials, this time showing the region in coefficient space where all roots have absolute value less than 2.
alt text http://dl.dropbox.com/u/5390048/CubicRootsSmaller2.jpg
Among these are 31 Perron polynomials (where the maximum is attained for a positive real root. Here are their roots, and the normalized roots (divided by the Perron number):
alt text http://dl.dropbox.com/u/5390048/PerronPoints3%282%29.jpg
alt text http://dl.dropbox.com/u/5390048/PerronPoints3%282%29normalized.jpg
After seeing and thinking about these pictures, it became clear that for polynomials
with roots bounded by $C > 1$, as the degree grows large, the volume in coefficient space
grows large quite quickly with degree, and appears to high volume/(area of boundary) ratio. The typical coefficients become large, and
most of the roots seem to change slowly as the coefficients change, so you don't
bump into the boundary too easily. If so, then
to get a random lattice point within this volume, it should work fairly well to first
find a random point chosen uniformly in coefficient space, and then move to the
nearest lattice point.
With that in mind, I tried to guess the distribution of roots (invariant by complex
conjugation), choose a random sample of $d$ elements chosen independently from this distribution, generate the polynomial with real coefficients having those roots, round
off the coefficients to the nearest integer, and looking at the resulting roots.
To my surprise, many of the roots were not very stable: the nearest integral polynomial
usually ended up with roots fairly far out of bounds, for any parameter values of
several distributions I tried. (Note: one constraint on the distribution is that the
geometric mean of absolute values must be an integer $\ge 1$. This rules out the
uniform distribution at least for small values of $C$).
After thinking harder about the stability question for roots (as the coefficients are
perturbed), I realized the importance of the interactions of nearby roots. Whenever
there is a double root, the roots move quickly when coefficients are changed --- i.e.,
the ratio of volume in coefficient space to volume in root space is relatively small.
It's as if nearby roots in effect have a repulsive force. The joint distribution of roots is important: you get wrong answers if you treat them as independent.
With this in mind, I tried an experiment where I clicked on a diagram to put in roots
for a controlling real polynomial by hand, while the computer found the roots of the nearest polynomial
with integer coefficients. With a little practice, this worked well. New roots "prefer" to
be inserted where the existing polynomial is high, so I shaded the diagram by absolute value of the polynomial, to indicate
good places for a new root. Sometimes, roots of the controlling polynomial become disassociated from roots of the nearest integer polynomial, and the result is often an out-of-bounds root not near any controlling root. In that case, deleting control roots that are disassociated brings it back into line. As the control roots are moved around, the algebraic integers jump in discrete steps, and these steps are much smaller when the control root distribution is in a good region of the parameter space.
Here's a screen shot from the experiment, (which is fun!):
alt text http://dl.dropbox.com/u/5390048/ControlRoots.jpg
Here, the convention is that each control point above the real axis is duplicated with its complex conjugate.
Each control point below the real axis is projected to the real axis, and gives a real
root for the control polynomial. All the control roots are shown in black, and the roots of the nearest integer polynomial are shown in red. For these positions, the red roots are nicely associated with black roots. It is a Perron polynomial, because a
real root has been dragged so that it has maximum modulus.
In the next screenshot, I have dragged several control roots into a cluster around
11 o'clock. The red roots weren't happy there, so they disassociated from the control roots
and scattered off in different directions, one of them out to a much larger radius. This is a good indication that
the ratio of coefficient-space volume to root-space volume is small. This polynomial is not
Perron.
alt text http://dl.dropbox.com/u/5390048/RootPerturb-disassociated.jpg
This experiment is much trickier for $C$ close to $1$: the coefficients are much smaller for a given degree, which makes the roots much less stable. They become more stable when there
are lots of roots spread out fairly evenly, mostly near the outer boundary.
Here is one method that in principle should give a nearly uniformly-random choice of
a polynomial with roots bounded by $C$, and I think, by approximating with the nearest
polynomial having integer coefficients, give a nearly uniform choice of algebraic integer
whose conjugates are bounded by $C$: Start from any polynomial whose roots are bounded by
$C$, for instance, a cyclotomic polynomial. Choose a random vector in coefficient space,
and follow a $C^1$ curve whose tangent vector evolves by Brownian motion on the unit sphere.
Whenever a root hits the circle of radius $C$, choose a new random direction in which
the root decreases in absolute value (i.e, use diffuse scattering on the surfaces).
The distribution of positions should converge to the uniform distribution in the
given region in coefficient space.
This method should also probably work to find a random polynomial whose roots are inside
any connected open set, and subject to certain geometric limitations (for instance, it can't be
inside the unit circle) a nearly uniformly random algebraic integer of high degree
whose Galois conjugates are inside a given connected open set.
Of course still more interesting than an empirical simulation would be a
good theoretical analysis.
Best Answer
This is a non-centered iid random matrix whose entries have mean one and variance one (and decay exponentially at infinity), and as such, is subject to the circular law with one outlier. Thus, there will be one eigenvalue roughly near n, and the rest will be more or less uniformly distributed in the complex disk of radius $\sqrt{n}$. The latter result is due to Chafai (at least for almost all of the eigenvalues), and the former is due to Silverstein. I discuss some finer aspects of the outlier eigenvalue (and show that all the other eigenvalues are nearly contained in the disk) in a more recent paper. (See in particular Phillip Wood's Figure 3 in that paper for an example of the eigenvalue distribution of a similar matrix model to the one you propose.)