I don't quite understand your eqs. 1 and 2 so I will focus on answering questions 4 and 5. Maybe someone else can provide more specific answers for your 1--3. But I'll point you towards a great reference for various test statistics here, which you might find helpful.
From what I can glean, you have (separately) generated samples of background-only and signal-only events, the latter for a collection of points in its 2-dimensional parameter space. You have then performed an analysis on a binned histogram of some final state invariant mass, using information obtained from similarly-binned histograms constructed from the signal and background event samples.
I place an emphasis on "binned" because it affects the likelihood function you define: take a background-only assumption, for example. Recall that the likelihood function is defined as the probability density function (PDF) generating the data, except with random variables interpreted as parameters and vice versa. For a binned histogram, the number of events falling within a given bin $k$ can be modelled as a Poisson distribution, and so: \begin{equation} L^\mathrm{(bckg)}_k(b_k;n_k) = P(n_k;b_k) = \frac{(b_k)^{n_k} e^{-b_k}}{n_k!}\,, \end{equation} with $n_k$ the number of events in the bin, and $b_k$ the expected number of background events. Treating bins as independent counting experiments, the full likelihood can then be written as the product over all bin likelihoods: \begin{equation} L^\mathrm{(bckg)}(b,n) = \prod_k L^\mathrm{(bckg)}_k(b_k;n_k)\,. 915
\end{equation} For the signal+background assumption, simply write $b_k\rightarrow b_k+s_k$. To address the first part of your q.$~$5, note that eq.$~$(86) in the text you linked defines likelihood functions for unbinned datasets (so-called "extended likelihoods"), so they do not apply to your analysis. (You can check that they do not allude to any binning.)
Since $s_k$ and $b_k$ are supposed to represent the expected number of signal or background events in bin $k$, they must clearly be related to the physics models of the signal and background, respectively. If we denote by $\mathcal{S}(m_{||};M,\theta)$ and $\mathcal B(m_{||};\Theta)$ the PDFs describing the signal-only and background-only invariant mass distributions, then for bin $k$ one has: \begin{equation} s_k=N_\mathrm{total}\times\frac{\int_{m_{k_\mathrm{min}}}^{m_{k_\mathrm{max}}} \mathcal S(m_{||};M,\theta)\,dm_{||}}{\int_{0}^{\infty} \mathcal P_\mathrm{total}(m_{||})\,dm_{||}}\,, \end{equation} where $m_{k_\mathrm{min}}$ and $m_{k_\mathrm{max}}$ are the invariant mass bounds for bin $k$, and $N_\mathrm{total}$ is the total number of observed events. $\mathcal P_\mathrm{total}(m_{||})$ is the total PDF; if assuming signal+background for example, then $\mathcal P_\mathrm{total}=\mathcal S+\mathcal B$. (Of course, if the total PDF is properly normalised then the denominator is 1.) This is simply the fraction of (signal) events you expect to fall within the $k^\mathrm{th}$ bin, given the theory PDF. $b_k$ can be written analogously.
However, you bypass the need to include such calculations for your likelihoods because you instead construct signal-only and background-only histograms from generated events, and directly obtain $s_k$ and $b_k$ from there. (This addresses your q.$~$4 and the second part of your q.$~$5.) 818