I am going to use the following results :
$$V_1=\sum_{n=1}^\infty\frac{(-1)^nH_n^{(4)}}{n}=\frac78\ln2\zeta(4)+\frac38\zeta(2)\zeta(3)-2\zeta(5)$$
$$V_2=\int_0^{1/2}\frac{\ln^3(1-x)}{x(1-x)}\ dx=6\operatorname{Li}_4\left(\frac12\right)-6\zeta(4)+\frac{21}{4}\ln2\zeta(3)-\frac32\ln^22\zeta(2)+\frac14\ln^42$$
$$V_3=\int_0^{1/2}\frac{\ln^3(1-x)\ln x}{x(1-x)} dx=-12\operatorname{Li}_5\left(\frac12\right)-12\ln2\operatorname{Li}_4\left(\frac12\right)+\frac{285}{16}\zeta(5)-3\zeta(2)\zeta(3)$$
$$-\frac{21}{4}\ln^22\zeta(3)+2\ln^32\zeta(2)-\frac25\ln^52$$
$$V_4=\sum_{n=1}^\infty\frac{H_n^{(3)}}{n^22^n}=4\operatorname{Li}_5\left(\frac12\right)+3\ln2\operatorname{Li}_4\left(\frac12\right)-\frac{81}{64}\zeta(5)+\frac{5}{16}\ln2\zeta(4)-\frac78\zeta(2)\zeta(3)$$
$$+\frac78\ln^22\zeta(3)-\frac{5}{12}\ln^32\zeta(2)+\frac{11}{120}\ln^52$$
We are going to establish two relations and solve them by eliminations
The first relation:
From here we have
$$-\int_0^1x^{n-1}\ln^3(1-x)\ dx=\frac{H_n^3+3H_nH_n^{(2)}+2H_n^{(3)}}{n}$$
Multiply both sides by $\frac{1}{n2^n}$ then sum both sides from $n=1$ to $\infty$ we get
$$\sum_{n=1}^\infty\frac{H_n^3+3H_nH_n^{(2)}+2H_n^{(3)}}{n^22^n}=-\int_0^1\frac{\ln^3(1-x)}{x}\sum_{n=1}^\infty\frac{(x/2)^n}{n}=\int_0^1\frac{\ln^3(1-x)\ln(1-x/2)}{x}\ dx\\\overset{x\mapsto 1-x}{=}\int_0^1\frac{\ln^3x\ln\left(\frac{1+x}{2}\right)}{1-x}\ dx=-6\sum_{n=1}^\infty\frac{(-1)^nH_n^{(4)}}{n}=-6V_1$$
Note that for the last step, we used the rule $$\int_0^1\frac{\ln^a x\ln\left(\frac{1+x}{2}\right)}{1-x}\ dx=(-1)^a a!\sum_{n=1}^\infty\frac{(-1)^nH_n^{(a+1)}}{n}$$
Then
$$R_1=\sum_{n=1}^\infty\frac{H_n^3+3H_nH_n^{(2)}+2H_n^{(3)}}{n^22^n}=-6V_1$$
The second relation:
From here we have
$$-\frac{\ln^3(1-x)}{1-x}=\sum_{n=1}^\infty x^n\left(H_n^3-3H_nH_n^{(2)}+2H_n^{(3)}\right)\tag{1}$$
Divide both sides of (1) by $x$ then integrate from $x=0$ to $1/2$ and use the fact that $\int_0^{1/2}x^{n-1}\ dx=\frac1{n2^n}$ to get
$$\sum_{n=1}^\infty\frac{H_n^3-3H_nH_n^{(2)}+2H_n^{(3)}}{n2^n}=-\int_0^{1/2}\frac{\ln^3(1-x)}{x(1-x)}\ dx=-V_2$$
Now multiply both sides of (1) by $-\frac{\ln x}{x}$ then integrate from $x=0$ to $1/2$ and use that fact that $-\int_0^{1/2}x^{n-1}\ln x\ dx=\frac{\ln2}{n2^n}+\frac1{n^22^n}$ to get
$$\sum_{n=1}^\infty\left(H_n^3-3H_nH_n^{(2)}+2H_n^{(3)}\right)\left(\frac{\ln2}{n2^n}+\frac1{n^22^n}\right)=\int_0^{1/2}\frac{\ln^3(1-x)\ln x}{x(1-x)}\ dx=V_3$$
$$-\ln2V_2+\sum_{n=1}^\infty\frac{H_n^3-3H_nH_n^{(2)}+2H_n^{(3)}}{n^22^n}=V_3$$
Then
$$R_2=\sum_{n=1}^\infty\frac{H_n^3-3H_nH_n^{(2)}+2H_n^{(3)}}{n^22^n}=V_3+\ln2V_2$$
Now we are ready to calculate our sums and lets start with the first one:
$$\sum_{n=1}^\infty\frac{H_nH_n^{(2)}}{n^22^n}=\frac{R_1-R_2}{6}=-\frac{6V_1+\ln2V_2+V_3}{6}$$
and by plugging the results of $V_1$, $V_2$ and $V_3$ we get
\begin{align}
\sum_{n=1}^\infty\frac{H_nH_n^{(2)}}{n^22^n}&=2\operatorname{Li}_5\left(\frac12\right)+\ln2\operatorname{Li}_4\left(\frac12\right)-\frac{31}{32}\zeta(5)+\frac{1}{8}\ln2\zeta(4)+\frac18\zeta(2)\zeta(3)\\&\quad-\frac{1}{12}\ln^32\zeta(2)+\frac{1}{40}\ln^52
\end{align}
As for the second sum :
$$\sum_{n^1}^\infty\frac{H_n^3}{n^22^n}=\frac{R_1+R_2}{2}=\frac{-6V_1+\ln2V_2+V_3-4V_4}{2}$$
and by plugging the results of $V_1$, $V_2$, $V_3$ and $V_4$ we get
\begin{align}
\sum_{n=1}^\infty\frac{H_n^3}{n^22^n}&=-14\operatorname{Li}_5\left(\frac12\right)-9\ln2\operatorname{Li}_4\left(\frac12\right)+\frac{279}{16}\zeta(5)-\frac{25}{4}\ln2\zeta(4)-\frac78\zeta(2)\zeta(3)\\&\quad-\frac74\ln^22\zeta(3)+\frac{13}{12}\ln^32\zeta(2)-\frac{31}{120}\ln^52
\end{align}
.
Proofs of the results:
$V_1$ can be found here and $V_2$ can be found using wolfram but here is the steps
\begin{align}
V_2&=\int_0^{1/2}\frac{\ln^3(1-x)}{x(1-x)}\ dx\overset{x\mapsto 1-x}{=}\int_{1/2}^1\frac{\ln^3x}{x(1-x)}\ dx\\
&=\int_{1/2}^1\frac{\ln^3x}{x}\ dx+\int_{1/2}^1\frac{\ln^3x}{1-x}\ dx\\
&=-\frac14\ln^42+\sum_{n=1}^\infty \int_{1/2}^1x^{n-1}\ln^3x\ dx, \quad \text{apply integration by parts}\\
&=-\frac14\ln^42+\sum_{n=1}^\infty\left(\frac{\ln^32}{n2^n}+\frac{3\ln^22}{n^22^n}+\frac{6\ln2}{n^32^n}+\frac{6}{n^42^n}-\frac{6}{n^4}\right)\\
&=-\frac14\ln^42+\ln^42+3\ln^22\operatorname{Li}_2\left(\frac12\right)+6\ln2\operatorname{Li}_3\left(\frac12\right)+6\operatorname{Li}_4\left(\frac12\right)-6\zeta(4)\\
&\boxed{=6\operatorname{Li}_4\left(\frac12\right)-6\zeta(4)+\frac{21}{4}\ln2\zeta(3)-\frac32\ln^22\zeta(2)+\frac14\ln^42}
\end{align}
where we used $\operatorname{Li}_2\left(\frac12\right)=\frac12\zeta(2)-\frac12\ln^22$ and $\operatorname{Li}_3\left(\frac12\right)=\frac78\zeta(3)-\frac12\ln2\zeta(2)+\frac16\ln^32$
\begin{align}
V_3&=\int_0^{1/2}\frac{\ln^3(1-x)\ln x}{x(1-x)}\ dx\overset{x\mapsto 1-x}{=}\int_{1/2}^{1}\frac{\ln^3x\ln(1-x)}{x(1-x)}\ dx\\
&=-\sum_{n=1}^\infty H_n\int_{1/2}^1 x^{n-1}\ln^3x\ dx\\
&=-\sum_{n=1}^\infty H_n\left(\frac{\ln^32}{n2^n}+\frac{3\ln^22}{n^22^n}+\frac{6\ln2}{n^32^n}+\frac{6}{n^42^n}-\frac{6}{n^4}\right)\\
&=-\ln^32\sum_{n=1}^\infty\frac{H_n}{n2^n}-3\ln^22\sum_{n=1}^\infty\frac{H_n}{n^22^n}-6\left(\color{blue}{\ln2\sum_{n=1}^\infty\frac{H_n}{n^32^n}+\sum_{n=1}^\infty\frac{H_n}{n^42^n}}\right)+6\sum_{n=1}^\infty\frac{H_n}{n^4}
\end{align}
I managed here to prove
$$\color{blue}{\ln2\sum_{n=1}^\infty\frac{H_n}{n^32^n}+\sum_{n=1}^\infty\frac{H_n}{n^42^n}}=-\frac12\ln^22\sum_{n=1}^\infty\frac{H_n}{n^22^n}-\frac16\ln^32\sum_{n=1}^\infty\frac{H_n}{n2^n}+\frac12\sum_{n=1}^\infty\frac{H_n}{n^4}$$
$$-\frac{47}{32}\zeta(5)+\frac1{15}\ln^52+\frac13\ln^32\operatorname{Li}_2\left(\frac12\right)+\ln^22\operatorname{Li}_3\left(\frac12\right)+2\ln2\operatorname{Li}_4\left(\frac12\right)+2\operatorname{Li}_5\left(\frac12\right)$$
and $V_3$ simplifies into
$$\small{V_3=3\sum_{n=1}^\infty\frac{H_n}{n^4}+\frac{141}{16}\zeta(5)-\frac25\ln^52-2\ln^32\operatorname{Li}_2\left(\frac12\right)-6\ln^22\operatorname{Li}_3\left(\frac12\right)-12\operatorname{Li}_4\left(\frac12\right)-12\operatorname{Li}_5\left(\frac12\right)}
\\ \small{\boxed{=-12\operatorname{Li}_5\left(\frac12\right)-12\ln2\operatorname{Li}_4\left(\frac12\right)+\frac{285}{16}\zeta(5)-3\zeta(2)\zeta(3)-\frac{21}{4}\ln^22\zeta(3)+2\ln^32\zeta(2)-\frac25\ln^52\quad}}$$
where we substituted $\sum_{n=1}^\infty\frac{H_n}{n^4}=3\zeta(5)-\zeta(2)\zeta(3)$ along with the values $\operatorname{Li}_2\left(\frac12\right)$ and $\operatorname{Li}_3\left(\frac12\right)$.
To prove the last result of $V_4$, we use this generating function
$$\sum_{n=1}^\infty \frac{H_n^{(3)}x^n}{n}=\operatorname{Li}_4(x)-\ln(1-x)\operatorname{Li}_3(x)-\frac12\operatorname{Li}^2_2(x)$$
Divide both sides by $x$ then integrate from $x=0$ to $1/2$ and use the fact that $\int_0^{1/2} x^{n-1}\ dx=\frac1{n2^n}$ we get
\begin{align}
V_4&=\int_0^{1/2}\frac{\operatorname{Li}_4(x)}{x}\ dx-\underbrace{\int_0^{1/2}\frac{\ln(1-x)\operatorname{Li}_3(x)}{x}\ dx}_{IBP}-\frac12\int_0^{1/2}\frac{\operatorname{Li}^2_2(x)}{x}\ dx\\
&=\operatorname{Li}_5\left(\frac12\right)+\operatorname{Li}_2\left(\frac12\right)\operatorname{Li}_3\left(\frac12\right)-\frac32\int_0^{1/2}\frac{\operatorname{Li}^2_2(x)}{x}\ dx
\end{align}
@Song proved in this solution
$$\int_0^{1/2}\frac{\left(\operatorname{Li}_2(x)\right)^2}{x}\ dx=\frac12\ln^32\zeta(2)-\frac78\ln^22\zeta(3)-\frac58\ln2\zeta(4)+\frac{27}{32}\zeta(5)+\frac78\zeta(2)\zeta(3)\\-\frac{7}{60}\ln^52-2\ln2\operatorname{Li}_4\left(\frac12\right)-2\operatorname{Li}_5\left(\frac12\right)$$
by plugging this result along with the values of $\operatorname{Li}_2(1/2)$ and $\operatorname{Li}_3(1/2)$ the closed form of $V_4$ follows.
