Typical Growth Model Social Planner Problem
Q1: Your bullet point (b) is not how we usually interpret capital, it is usually not assumed that $k_0$ is divided between $k_0^p$, and $k_0^s$ the capital we use to produce and the capital we store for next period. Instead, we assume all the capital is used for production, but using it for production does not delete it,  it only depreciates it. For example, if you use your machines to produce pizza that does not mean that they are obsolete to be used next period. 

If you want you could add to the model that un-used capital does not get depreciated, but it is an easy exercise to show that under very general conditions it will never be optimal to leave capital un-used. 

So I do not agree with your interpretation in (e), typically $k_1=(1-\delta)k_0+i_0$. 

Q2: This question is not particular of the dynamic model you present, but I think that the infinite sequences are messing up with your intuition. 

Consider this simple example. If I say that you can choose to buy chocolate (at a price of \$2) or cheese (at a price of \$4), but you only have \$10 dollars (sorry this are rough times :P ), it is clear that if you are to spend all your money on these two goods, then choosing how much chocolate to buy, immediately pins down how much cheese you will buy. However, the fact that this is true does not change the nature that both amounts of chocolate and cheese are your choice variables. The budget constraint may allow you to re-formulate this problem as one where you only choose how much chocolate to buy (since cheese would be implicit), but this will only be a technique to simplify the original problem. 

Something similar happens here. So to be concise, both sequences $\{c_t,k_{t+1}\}_{t=0}^\infty$ are control variables even if they have a simple linear relationship between them through the law of motion.