Making edits to this question. I think I may have figured out how to do the first part of the question
Based on this website (http://www.stat.rice.edu/~dobelman/notes_papers/math/Riemann.Stiltjes.pdf) Using a combination of the different ways of solving the problem. I was able to get the right answer. I think I solved/reasoned for the answer correctly.
$\int_{-1}^{100} f(x)d\alpha(x)= x^2(\alpha(2+)- x^2(\alpha(2-) + x^2(\alpha(3+)- x^2(\alpha(3-) = x^2(\frac 23 x -1 - 0) + x^2(1-\frac 23 x -1 - 0)$
$\int_{-1}^{100} f(x)d\alpha(x)= \int_{-1}^{100}x^2d\alpha=f(100)\alpha(100)-f(-1)\alpha(-1)-\int_{-1}^{100} \alpha(n)2xdx$
$f(100)=100^2$
$\alpha(100)=1$
$f(-1)=(-1)^2$
$\alpha(-1)=0$
$(100)^2-0-\int_{-1}^{100} \alpha(n)2xdx$
$(100)^2-\int_{2}^{3} (\frac 23 x -1)*2x dx - \int_{3}^{100} \alpha(100)2x dx$
$\alpha(100)=1$
$(100)^2-\int_{2}^{3} (\frac 23 x -1)*2x dx - \int_{3}^{100} 2x dx$
Solving the integral with the limits gives the answer $\frac {50}{9}$
Still need help with the second part of the question. Since f is not given, would you just disregard it? I mean just use the give \alpha to solve the problem?
When $\alpha$ is constant, the Stieljes integral is $0$. Suppose $\alpha(x) = m$ everywhere on $[c,d]$. If $P=(\{x_i\}_{i=0}^n, \{t_i\}_{i=1}^n)$ is a partition of $[c,d]$, then the Riemann-Stieljes sum over $P$ for $\int_c^d f\,d\alpha$ is $$S(P,f) = \sum_{k=1}^n f(t_k)(\alpha(x_k) - \alpha(x_{k-1}))$$ But since $\alpha$ is constant, $\alpha(x_k) - \alpha(x_{k-1}) = 0$ for every $k$,so $S(P,f) = 0$, regardless of the partition $P$ or integrand $f$. Thus every function $f$ is integrable with respect to $\alpha$, and $$\int_c^d f(x)\,d\alpha(x) = 0$$
So in the case of your first problem,
$$\begin{align}\int_{-1}^{100}f(x)\,d\alpha(x) &= \lim_{\epsilon\to 0+}\int_{-1}^{2-\epsilon}f(x)\,d\alpha(x) + \int_{2-\epsilon}^{3}f(x)\,d\alpha(x) + \int_{3}^{100}f(x)\,d\alpha(x)\\ &= 0 + \int_{2-}^{3}f(x)\,d\alpha(x) + 0\end{align}$$ Where the $2-$ indicates that you still have to account for the discontinuity of $\alpha$ at $2$ within the remaining integral.
In the second problem, they are asking you to calculate $$\int_0^n x\,d\alpha(x)$$ The function that needs to be integrated is whatever is between the $\int$ and the $d\alpha(x)$ (as used here - there is another notational convention where the integrand is given after the $d\alpha(x)$ and you have to guess what is included in it and what is not). The integrand does not have to named "$f$". And the function with respect to which you are integrating does not have to named "$\alpha$", though they still did so here.
So you integrate the function $x$ with respect to the function $\lfloor x \rfloor$.