Definition. A family $\mathcal{F}\subseteq\mathcal{P}(\omega)$ has the finite intersection property if $\bigcap\sigma$ is infinite for all $\sigma\in[\mathcal{F}]^{<\omega}$.
Definition. An $A$-triple to be a triple of the form $(\mathcal{T},A,Q)$ such that the following conditions are satisfied.
1) $\mathcal{T}$ is a strong Choquet, second-countable topology on $2^{\omega}$ that is finer than the standard topology (cones topology).
2) $A\in\mathcal{T}$.
3) $Q$ is a non-empty countable subset of $A$ with no isolated points in the subspace topology it inherits from $\mathcal{T}$.
I am reading the following Lemma and I found two details that I don't see very clearly.
1) Reviewing the proof, I don't understand how $p'\in D_{\ell}^{ref}$, precisely, I'm not sure how to show that the collection of open $\{U_{s}^{p'}:s\in{^{n_{p'}}2}\}=\{U_{s}^{p'}:s\in{^{n_{p}+1}2}\}$ it's a cover of $2^{\omega}$.
2) Again, in Lemma 30, my idea to prove that $\mathcal{F}\cup\{\bigcap P\}$ has the property of finite intersection is to assume otherwise. So, since $\mathcal{F}$ does have the property of finite intersection, there is $\sigma=\{x_{1},...,x_{k}\}\in [\mathcal{F}]^{<\omega}$ such that \begin{align*} H=\left(\displaystyle{\bigcap_{i = 1}^{k}x_{i}}\right)\cap\left(\bigcap P\right) \end{align*} then $|H|<\omega$. Since $H$ is finite, let $\ell'=maxH$. Let's take $\ell=\ell'+1$ and consider the dense set $D_{\sigma,\ell}$. Since $G$ is a $\mathcal{D}$-generic filter, there is $p\in G\cap D_{\sigma,\ell}$. If $x=\bigcap P$ then I must show that there is $s\in{^{n_{p}}2}$ such that $x\in U_{s}^{p}$ because there will be $i>\ell$ such that $i\in\left(\displaystyle{\bigcap_{i = 1}^{k}x_{i}}\right)\cap\left(\bigcap P\right)$ by the definition of $D_{\sigma,\ell}$ contradicting that $\ell'$ is the maximum of $H$. But I could not do it.
Any hint will help me a lot.
Concerning your first question, I believe that they are only using “refines” in the sense of being just finer, not a finer partition. Perusing the second picture (p. 16) the density of $D_{\ell}^{\mathrm{ref}}$ seems to be only needed to show that $\bigcap_{n \in \omega} U_{x | n}$ is a singleton.
I'll address the second question later.