Definition of a cosemisimple Hopf algebra A cosemisimple Hopf algebra is usually defined to a Hopf algebra which is the sum of its cosimple subcoalgebras. Does this definition assume that each  cosimple subcoalgebra appears only once in the sum, or is this a consequence?
Also, assuming this is true, then the decomposition must be unique by a cosimplicity argument, i.e. given two decompositions $H = \bigoplus_i H_i = \bigoplus H'_i$, with each $H_i$ and $H'_i$ cosemisimple, then for each $i$ there exists a $j$ such that $H_i = H'_j$.
 A: For the finite dimensional case, cosemisimple is the same as the dual being semisimple.  So you can apply your knowledge of semisimple algebras.  So the answer to your first question is no, you're allowed to have repeats just like semisimple algebras can have the same matrix factor repeated.
For uniqueness of decomposition, look at H as a comodule over itself, and use isotypic decomposition.  This gives exactly the uniqueness statement you want.
A: The answer to your first question is in general negative. Another way to see this (apart from the one indicated in Noah Snyder's answer above) is to consider a counterexample: Take the case of a group $G$ and its group hopf algebra $kG$. Then it can be shown that 
$$
corad(kG)=\bigoplus_{g\in G}kg\cong kG
$$
where $corad$ stands for the coradical of $kG$ (i.e. the sum of all its simple subcoalgebras). The above isomorphism should be interpreted as an isomorphism of $k$-coalgebras. Thus, the group hopf algebra is cosemisimple as a coalgebra (thus, it is by definition cosemisimple as a Hopf algebra) and pointed. Also $kg$, for all $g\in G$ is simple (since it is $1$-dim) and $kg_1\cong kg_2$ (as $k$-coalgebras) for all $g_1,g_2\in G$. 
As for your second question, the argument already provided in user's Noah Snyder answer is neat. 
