Reduction of algebraic groups Let $G$ be an algebraic group over $\mathbb{Z}_p$ embedded in $GL_n$. Let's say that $G$ is a family of equation $f \in \mathbb{Z}_p[(X_{i,j})_{i,j}]$ such that the set of invertible matrices satisfying the equations form a subgroup.

Is the natural projection $G(\mathbb{Z}_p) \rightarrow G(\mathbb{F}_p)$ surjective ? If not, is there a criterion ?

For example it is true for $G=GL_n$ and $SL_n$. Note that $GL_n(\mathbb{Z}) \rightarrow GL_n(\mathbb{Z}/ N \mathbb{Z})$ is not surjective. For $G=Sp_n$, $O_n$, $SO_n$... I don't know.
 A: The answer is "no" in the generality you pose the question, but "yes" for every example you ask about.
Dumb counter-example: Inside $GL_1$, consider $\{ (x) : p(x-1)=0 \}$. Inside $\mathbb{Z}_p$, the only solution is the identity matrix, but in $\mathbb{F}_p$ every matrix is a solution. Also note that $p(x^{-1}-1) = - x^{-1} p (x-1)$ and $p (xy-1) = p (x-1) + xp (y-1)$. So, for any commutative ring $R$, the set of invertible elements obeying $p(x-1)=0$ is a group under multiplication. This shows that this equation defines a sub-group-scheme of $GL_1$.
This example is not interesting because we imposed the equation $pf=0$ without imposing $f=0$. It is natural to only study sub-group-schemes of $GL_n$ over $\mathbb{Z}_p$ where, if $pf$ is in the defining ideal, then so is $f$. The technical term is that we are only interested in flat sub-group-schemes of $GL_n$.
A flat counterexample Consider the subgroup $G$ of $SL_2$
$$\left\{ \begin{pmatrix} a & b \\ b & a \end{pmatrix} : a^2-b^2 = 1 \right\}.$$
That's a hypersurface in $\mathbb{A}^2_{\mathbb{Z}}$ whose defining equation is not divisible by any prime, so it is flat. But the point $\left( \begin{smallmatrix} 0 & 1 \\ 1 & 0 \end{smallmatrix} \right) \in G(\mathbb{Z}/2 \mathbb{Z})$ doesn't lift to $G(\mathbb{Z}/4 \mathbb{Z})$. 
Why all of your examples are fine There is a strengthening of Hensel's lemma which says: Suppose we have a system of equations with coefficients in $\mathbb{Z}_p$. Let the system be flat (if we set $pf=0$, we also set $f=0$) and let the solutions over $\mathbb{F}_p$ be smooth. (If you don't know what smooth means in finite characteristic, you should look in an algebraic geometry textbook.)
Then all $\mathbb{F}_p$ solutions to your equations lift to $\mathbb{Z}_p$. The classical examples you give, such as the orthogonal group, symplectic group etc, are all smooth over $\mathbb{F}_p$. (Maybe with some technical issues about getting the definition right at the prime $2$.)
