Let $\bf{G}$ be a simple and simply connected algebraic group over $\mathbb{Q}$. Is it true that the base extension of $\bf{G}$ over $\bar{\mathbb{F}}\_p$, i.e., ${\bf G}\times\_{\mathrm{Spec(\mathbb{Z})}}\mathrm{Spec}(\bar{\mathbb{F}}\_p)$, is simple and simply connected for all but finitely many p? 



base extension of algebraic groups
The answer to the "reasonable" interpretation of your question is "yes". The real point is that the property of being "simply connected" in the sense of connected semisimple groups is characterized in terms of the root datum of the geometric fiber, and there is a "local constancy" property for root data in the setting of reductive group schemes over any ring, ultimately based on considerations with tori and Lie algebras over rings. 


(**Warning**: "simply connected" for connected semisimple groups in positive characteristic has *nothing* to do with the triviality of the etale fundamental group, as the latter is *always* nontrivial for smooth affine varieties of positive dimension in positive characteristic. Rather, it is defined in terms of the absence of nontrivial central isogenous covers by other connected semisimple groups, and is very easy to verify for SL$\_n$ directly. It happens to be equivalent to triviality of the etale fundamental group in characteristic 0, but that is an entirely separate matter of no logical relevance to the question about properties in positive characteristic.)


The discussion addresses this from a broader perspective. If all you are interested in is SL$\_n$ or any other "simply connected classical group" then nothing below is needed: all of the "classical groups" can be treated directly over any algebraically closed field on its own terms by directly checking that the coroots span the cocharacter lattice of a maximal torus, from which "simply connected" is immediate via how it is *defined* in a characteristic-free manner. Likewise for being "simple": it is a standard exercise for the classical groups, and is also characterized by irreducibility of the root system (which can be computed in each classical case).


Now for the general perspective. 
By easy "denominator-chasing" stuff, if $A$ is a domain with fraction field $K$ and $G$ is an affine $K$-group scheme of finite type then there exists a nonzero $a \in A$ and an affine $A[1/a]$-group scheme $\mathbf{G}$ of finite type with generic fiber $G$. Much more serious is that for a suitable nonzero multiple $a'$ of $a$ in $A$ (i.e., passing to a dense open subscheme ${\rm{Spec}}(A[1/a'])$ of ${\rm{Spec}}(A[1/a])$) various properties of the generic fiber as a $K$-scheme are satisfied by $\mathbf{G}\_{A[1/a']} \rightarrow {\rm{Spec}}(A[1/a'])$. 


Among such properties are "geometrically connected fibers" [EGA IV$\_3$, 9.7.7(ii)] (this is not obvious, as it is *false* for merely "connected fibers", but a connected scheme with a rational point over a field is automatically geometrically connected -- a simple exercise in the finite type case which is all you need, and [EGA IV$\_2$, 4.5.14] in general), flatness [EGA IV$\_3$, 11.2.6.1(ii)] (though for your situation over a Dedekind domain this much is trival since flat = torsion-free over a Dedekind base), and even smoothness (by tracking the Jacobian condition and fiber dimension in the presence of flatness, or by [EGA IV$\_4$, 17.7.8(ii)]). Actually, it is a general (not obvious) fact that a smooth relatively affine group scheme over an irreducible scheme has all fibers connected if the generic fiber is connected, but this is not needed and so we will pass over it in silence.


So by such general result (or hand-waving), it is "standard" upon that $G$ extends to a smooth affine $A[1/a]$-scheme $\mathbf{G}$ with (geometrically) connected fibers for some nonzero $a \in A$. With $A = \mathbf{Z}$ this says for any smooth connected affine $\mathbf{Q}$-group $G$ there exists a nonzero integer $N$ such that $G$ is the generic fiber of a smooth affine $\mathbf{Z}[1/N]$-group $\mathbf{G}$ such that $\mathbf{G}\_{\mathbf{F}\_p}$ is (geometrically) connected all all $p \nmid N$. 


Now comes the real content: if $G$ is reductive (resp. semisimple, resp. semisimple and simply connected) then does the same hold for $\mathbf{G}\_{\mathbf{F}\_p}$ for all but finitely many $p \nmid N$? This has no number-theoretic content at all, as it is really an assertion about group schemes over arbitrary domains: if $G$ is a smooth affine group scheme over a domain $A$ with fraction field $K$ such that all fibers are connected and if $G\_K$ is reductive (resp. semisimple, resp. semisimple and simply connected) then does the same hold for the fibers $G\_s$ over points $s$ in some dense open subscheme $U \subset {\rm{Spec}}(A)$? The answer is affirmative (and has nothing whatsoever to do with any classification theory for reductive or semisimple groups over fields, nor anything to do with the existence of Chevalley groups): for reductivity as well as for semisimplicity these are part of [SGA3, XIX, 2.6], and are also proved as Prop. 3.1.9(1) in the article "Reductive group schemes" in "Autour des schemas en groupes", no. 42-43 of Panoramas en Syntheses, SMF. One could instead make arguments via classification theorems and Chevalley groups. Choose your own poison.


Once that is done, and we arrange that $G$ is reductive over $A$, all geometric fibers have the *same* root datum (so if one fiber is simply connected then so are all fibers). This is elementary: we pick a maximal $K$-torus $T\_{\eta} \subset G\_K$ and a finite extension $K'/K$ such that $(T\_{\eta})\_{K'}$ is split. Then by replacing $A$ with some $A[1/a]$ we can find a domain $A' \subset K'$ finite flat over $A$ such that (i) $K' = {\rm{Frac}}(A')$ and (ii) $(T\_{\eta})\_{K'}$ "spreads out" to a closed $A'$-subgroup $T' \subset G\_{A'}$ that is a split fiberwise maximal torus in $G\_{A'}$. Since ${\rm{Spec}}(A') \rightarrow {\rm{Spec}}(A)$ is surjective, we can rename $A'$ as $A$ so that $G$ contains a closed $A$-subgroup $T$ which is a split fiberwise maximal torus. By localizing a bit more on $A$, the action of $T$ on ${\rm{Lie}}(G)$ then decomposes into a direct sum of ${\rm{Lie}}(T)$ and free modules of rank 1 on which $T$ acts through a fiberwise nontrivial character. In a similar manner we get the coroots as cocharacters of $T$ over $A$ (localizing a bit more on $A$, though that isn't really necessary) by using the intrinsic characterization of coroots. So this builds a root datum using the character and cocharacter groups of the split $A$-torus $T$, and as such identifies the root data of all fibers. Since "simple" for a connected semisimple group over an algebraically closed field is characterized by irreducibility of the root system, that property also spreads out from the geometric generic fiber. 


