1)
I applied Raabe's test on both and guessed the answer as (B), but not convinced enough. Is there a better approach?
2) Is there any closed form of following the series. It is however, known that the sum is irrational. (sorry for poor formatting)
Series $(ii)$ diverges, since the absolute value of its $n$th term converges to 1 rather than 0.
Series $(i)$ diverges as well. For $x$ small, $\sin(x)=x+o(x^2)$ (recall the Taylor series for $\sin(x)$ about 0), so the given series behaves like the harmonic series plus a convergent series.
There is no known closed form (in terms of elementary expressions) for the series. However, it can be expressed in terms of Jacobi's theta function as $\displaystyle\frac{\vartheta_3(0,1/2)-1}2$.
(A good idea when looking for closed form expressions for numerical series is to first try is the wonderful page for Plouffe's inverter.)