If $C$ is a graded coalgebra (e.g. $C$= the homology of a d.g. Hopf algebra), then $C_0$ is not necesarily a subcoalgebra, because
$$\Delta(C_0)\subset (C\otimes C)_0=\oplus_{n\in\mathbb Z}C_n\otimes C_{-n}$$ Think of a non-trivially graded finite dimensional algebra (for instance $M_n(k)$ with $|E_{ij}|=i-j$), then there are nonzero elements in $A_n\otimes A_{-n}$ that maps to non-zero elements in $A_0$. If $C=A^*$ then you have an example where $\Delta(C_0)$ has non-trivial components outside $C_0\otimes C_0$. (If you ar worried about signs, just take $C=\oplus_{n\in\mathbb Z}C_{2n}$ where $C_{2n}=A_n^*$.) (if you want a Hof algebra, examples are more dificult, but I think you now have the idea of why this hypohesis on the grading is needed)

Of course if $C=\oplus_{n\geq 0} C_n$, then $C_n=0$ for $n<0$ and $C_{-n}=0$ for $n>0$, so, the ony nonzero summand in $\oplus_{n\in\mathbb Z}C_n\otimes C_{-n}$ is $C_0\otimes C_0$.

If $C$ is a graded coalgebra (e.g. $C$= the homology of a d.g. Hopf algebra), then $C_0$ is not necesarily a subcoalgebra, because
$$\Delta(C_0)\subset (C\otimes C)_0=\oplus_{n\in\mathbb Z}C_n\otimes C_{-n}$$

For example, $H=k\{x,y\}/(x^2=y^2=xy+yx)$ is a graded Hopf algebra with $|x|=1$ and $|y|=-1$, both $x$ and $y$ primitives (in particular a d.g. Hopf algebra with $d=0$ and agree with its homology).

$H_0=k\oplus kx\wedge y$, and $$ \Delta(x\wedge y)= (x\otimes 1+1\otimes x)(y\otimes 1+1\otimes y)$$ $$= x\wedge y\otimes 1+1\otimes x\wedge y+ x\otimes y-y\otimes x \notin H_0\otimes H_0$$

Of course if $C=\oplus_{n\geq 0} C_n$, then $C_n=0$ for $n<0$ and $C_{-n}=0$ for $n>0$, so, the only nonzero summand in $\oplus_{n\in\mathbb Z}C_n\otimes C_{-n}$ is $C_0\otimes C_0$.