For the general question, assuming picks are without replacement:


for each numbered ball picked by C from their own urn, the probability C did not get that numbered ball from A is $\displaystyle \frac34$ (i.e. it went to B, D or E), 
so the probability of no matches is $\displaystyle \left(\frac34\right)^N$, 
the probability of at least one match is $\displaystyle 1-\left(\frac34\right)^N$ and 
the probability of $n$ matches is $\displaystyle {N \choose n}\frac{3^{N-n}}{4^N}$. 


For the scenarios the numbered balls received by C from A are determined by the scenario, so:


if C picks one from ten balls, the probability it is number $7$ is $\dfrac{1}{10}$
if C picks two from ten balls, the probability they are numbers $7$  and $8$ is $\dfrac{1}{45}$
if C picks two from ten balls, the probability exactly one is number $7$  or $8$ is $\dfrac{16}{45}$, and the probability at least one is number $7$  or $8$ is $\dfrac{17}{45}$ 
if C picks two from ten balls, the probability one is number $7$ is $\dfrac{1}{5}$ 
if C picks one from ten balls, the probability it is numbers $4$, $5$, $6$ or $7$ is $\dfrac{2}{5}$


Double urn probability and multiple instances
Is it true that using scratched Teflon-coated pans contain carcinogens, and if so, can they be consumed through the food cooked in them?

E.g.


The deadly toxins from non-stick frying pans

