I figured my comments are getting too long so I should put some of them here, even though I'm using a different topology on $\pi_1$ (often it is the same, but certainly not always) and a possibly different pro-group (which I can't tell as I'm not familiar with toposes).
EDIT: Unless otherwise noted, I'm assuming that $X$ is compact metrizable to be on the safe side.
The relationship goes in 2 steps. First the fundamental pro-group can be compared with the Cech fundamental group endowed with the inverse limit topology. In particular, as noticed by Atiyah and G. Segal, they contain precisely the same information as long as the fundamental pro-group is Mittag-Leffler. For other results in this direction see lemma 3.4 in "Steenrod homotopy". The topological homomorphism between $\pi_1(X)$ (with topology as in my comment) and the topological Cech fundamental pro-group is discussed in theorem 6.1, section 5 and elsewhere in "Steenrod homotopy".
To summarize the relationship very roughly, $\pi_1(X)$ topologized as in my comment retains much of the inverse limit of the fundamental pro-group, discards all of its derived limit, but instead gets something of the derived limit of the second homotopy pro-group (exactly how much is still a subject of ongoing research, see Theorem 6.5 and remark to corollary 8.8 in "Steenrod homotopy"). Still this is not all that it contains (cf. example 5.7 in "Steenrod homotopy").
Added later: Addressing Mike's comment, let me elaborate on the choice of topology on $\pi_1$.
The Mathoverflow thread on the quotient topology on $\pi_1$, and particularly Andrew Stacey's answer there, make it clear that the only reason that this topology is not compatible with multiplication is that the product of two quotient maps need not be a quotient map. But wait, the product of two quotient maps is a quotient map in the category of uniform spaces and uniformly continuous maps! See Isbell's "Uniform spaces" (1964), Exercise III.8(c). In fact, the product of any (possibly infinite) collection of quotient maps is a quotient map in the uniform category. (Quotient maps are defined in any concrete category over the category of sets, as explained e.g. in The Joy of Cats.)
This means that the topology of the quotient uniformity on $\pi_1(X)$ makes it into a topological group. By this topology I mean the following. If $X$ is compact, it carries a unique uniformity, and then the space of continuous (=uniformly continuous) maps $(S^1,pt)\to (X,pt)$ is a uniform space (endowed with the uniformity of uniform conergence; if $X$ is metrizable by a metric $d$, this uniformity is metrizable by the metric $D(f,g)=\sup\limits_{x\in X}\ d(f(x),g(x))$.)
If $X$ is not compact, then we need to fix some uniformity on it and the above works. To what extent the resulting topology on $\pi_1(X)$ depends on the choice of uniformity is a good question.
Added still later: With this in mind, I no longer see a good reason for myself to think at all about the quotient topology on $\pi_1(X)$. As you can see for instance from my comment here, I have good reasons to believe that quotient topology is a sensible notion only when it coincides with the topology of quotient uniformity (i.e. for quotients of compact spaces), and otherwise the topology of quotient uniformity is the way to go. The present situation does not seem to be an exception.
Now does the topology of the quotient uniformity coincide with the "inverse limit" topology, at least when $X$ is compact? I find it easier to go down one dimension, and compare the two similar topologies on $\pi_0(X)$, the set of path-components. The "inverse limit" topology is again induced from the inverse limit topology on the Cech $\pi_0$ (which is the inverse limit of $\pi_0(P_i)$, where $X$ is the inverse limit of the compact polyhedra $P_i$), and alternatively its basic open sets are those collections of path components whose unions are the point-inverses of maps of $X$ to finite sets. And as long as $X$ is compact (or is endowed with a uniformity), we have the topology of the quotient uniformity, viewing $\pi_0(X)$ as the quotient of $X$ by the equivalence relation of being in the same path-component.
Firstly let me say that (often or not) the two topologies do share something. For instance if $X$ is the $p$-adic solenoid then both are anti-discrete, for apparently very different reasons: the inverse limit topology simply because $X$ is connected; and the topology of the quotient uniformity is antidiscrete because every point lies in the closure of the path-component of every other point. If you take a wedge of two solenoids or even a "garland" of solenoids obtained from their disjoint union by identifying pairs of points taken from distincts solenoids, both topologies are still antidiscrete. In the case of the topology of the quotient uniformity, this is so because every two points $x,y$ are the endpoints of a chain $x=p_0,\dots,p_n=y$ such that each $p_{i+1}$ lies in the closure of the path-component of $p_i$.
On the other hand, let us do the pseudo-arc. Its path-components are single points, so $\pi_0(X)=X$. The topology of the quotient uniformity is the original topology of $X$; and the "inverse limit" topology is anti-discrete because $X$ is connected.
I haven't really thought about the two topologies on $\pi_1$ but I see no reason why this should be terribly different from the case of $\pi_0$.
Still more: While the relation of the two topologies on $\pi_1$ might be not so easy to comprehend, the topology of the quotient uniformity on $\pi_1$ relates in a seemingly more comprehensible way directly to the topologized Steenrod $\pi_1$ (whose relation to the fundamental pro-group is clear). The Steenrod $\pi_1(X)$ can be defined as
$\pi_1(holim P_i)$, where the compact metrizable space $X$ is the inverse limit of the polyhedra $P_i$ and $P_0=pt$. (If $P_0\ne pt$ we can shift the indices by one and add a new $P_0$, namely $pt$.) If you think about the construction of $holim$, you will see that $\pi_1(holim P_i)$ is the set of equivalence classes of level-preserving base-ray-preserving maps $S^1\times [0,\infty)\to P_{[0,\infty)}$ under the relation of level-preserving base-ray-preserving homotopy, where $P_{[0,\infty)}$ denotes the mapping telescope of the inverse sequence $\dots\to P_1\to P_0$ (glued out of the mapping cylinders $P_{[i,i+1]}$ of the individual bonding maps). Since $P_0=pt$, "level-preserving" can in fact be replaced by "proper" (see Lemma 2.5 in "Steenrod homotopy"). In other words, the Steenrod $\pi_1(X)$ is the set of equivalence classes of base-ray preserving maps from $D^2$ to the one-point compactification $P_{[0,\infty)}^+$ of the mapping telescope such that the preimage of the point at infinity is precisely the center of the disk (and nothing else).
Now it is not hard to see that the topology on the Steenrod $\pi_1(X)$ (induced from the inverse limit topology on the Cech $\pi_1(X)$ ) is precisely the topology of the quotient uniformity. The uniformity is on a subspace of the space of all continuous (=uniformly continuous) maps from $D^2$ to $P_{[0,\infty)}^+$, and so it is the subspace uniformity (of the uniformity of uniform convergence).
The continuous map from $\pi_1(X)$ with the topology of the quotient uniformity to the Steenrod $\pi_1(X)$ is given by Milnor's lemma (see Lemma 2.1 in Steenrod homotopy): every map $S^1\to X$ extends to a level-preserving map $S^1\times [0,\infty]\to P_{[0,\infty]}$, where $P_{[0,\infty]}$ is Milnor's compactification of the mapping telescope by a copy of $X$, and this extension is well-defined up to homotopy through such extensions. Also close maps have close extensions (from the proof of Milnor's lemma). The quotient $P_{[0,\infty]}/X$ is homeomorphic to $P_{[0,\infty]}^+$, so the close extensions descend to close representatives of elements of the Steenrod $\pi_1$.
Best Answer
This was answered in the comments. The answer to both questions is "Yes". As Fernando Muro mentioned, any countable group is the fundamental group of a manifold that can be embedded in $\mathbb{R}^5$.
There is an open subset of $\mathbb{R}^3$ with fundamental group $\mathbb{Q}$, the complement of a particular solenoid.
Let $C_1$ be the complement of an unknotted solid torus. $\pi_1(C_1) = \langle x_1 \rangle$. Consider an unknotted solid torus inside the first which wraps around it twice. Let the complement of this be $C_2$. There is an inclusion $\pi_1(C_1) \hookrightarrow \pi_1(C_2) = \langle x_2 \rangle$ so that $x_1 = x_2^2$, or in additive notation, $x_1 = 2x_2$.
Let the $n$th solid torus wrap around the inside of the $n-1$st solid torus $m_n=n$ times. Let $C_n$ be the complement of this solid torus. There is an inclusion $\pi_1(C_{n-1}) \hookrightarrow \pi_1(C_n) = \langle x_n \rangle$ so that $x_{n-1} = n x_n$.
The solenoid is the intersection of this infinite sequence of nested solid tori. The intersection of the solenoid with a meridianal disk of the first solid torus is a Cantor set. The solenoid can be viewed as a mapping cylinder of an automorphism of a Cantor set. The complement $C$ is the union of the complements. Any loop in the complement is contained in some $C_n$. The fundamental group is the direct limit, isomorphic to $\mathbb{Q}$ with $x_n = 1/n!$.
If you choose $m_n=2$ instead of $m_n=n$, the solenoid complement has a fundamental group isomorphic to the dyadic rationals. You can select other subgroups of $\mathbb{Q}$ by varying the sequence $( m_n )$.