So here are two questions:
Construct a holomorphic and bounded function on $\mathbb{D}$ with infinitely many zeros in $\mathbb{D}$ accumulating in $i$.
And construct and entire function with only zeros at $\{in^{\frac{1}{2}}\}, n \in \mathbb{N}$
For the first part I was going to go with Wierstrass Factorization theorem, but I believe the theorem only works if you do not have accumulation points. If so, may be something like a heavily modified $\tan(z)$ could work? I'm not sure, any help is appreciated.
The second part is this a good function? $f(z)=\prod^\infty_{n=0}(1-z/in^{\frac{1}{2}})$. Thanks!
$\sin(z)$ is bounded on the strip $-\pi/2 < \text{Im}(z) < \pi/2$ and has zeros at $\pi n$ for integers $n$. The strip can be mapped conformally to the right half plane $\mathbb H$ by $w = \exp(z)$, with $z = n \pi$ mapped to $w = \exp(n\pi)$ converging to $0$ as $n \to -\infty$, and then to the unit disk by $\zeta = 2i/(1+w)-i$ with $\zeta \to i$ as $w \to 0$. Inverting these maps, $z = \text{Log}(w) = \text{Log}\left(\frac{i-\zeta}{i+\zeta}\right)$ where $\text{Log}$ is the principal branch of the logarithm, so your function in the first part can be $$ \sin \left(\text{Log}\left(\frac{i-\zeta}{i+\zeta}\right)\right) $$
For the second part, you want to use the Weierstrass Factorization Theorem.