Can Stochastic Integration be Further Generalized? Is the idea of stochastic integration to accept convergence towards the stochastic integrals in probability instead of almost surely (pathwise)? 
I.e. to accept a weaker form of convergence for the Riemann sums in exchange for having a wider range of integrators?
(Since desired integrators, like Brownian motion, are almost surely of unbounded variation on any interval, convergence a.s. of the Riemann sums is impossible.)
Can the range of integrators be expanded even further beyond semimartingales if we ask only for convergence in distribution instead of convergence in probability?
(Since using Ito isometry we actually show convergence of the Riemann sums in L2, which implies their convergence in probability, I believe.)
 A: There exist different notions of stochastic integrals, some which apply integrators which are not semimartingales. In these cases, some regularity is usually lost. One primary example is integration with respect to fractional Brownian motion. A fractional Brownian motion $B_H$ with Hurst index $H$ satisfying $0<H<1$ is a continuous-time process with continuous paths and finite-dimensional distributions which are Gaussian. For $H = 1/2$, $B_H$ is a Brownian motion and thus a semimartingale. For $H\neq 1/2$, $B_H$ is not a semimartingale. There is a considerable litterature on stochastic integration with respect to fractional processes, fractional Brownian motion in particular. See for example the book "Stochastic Calculus for Fractional Brownian Motion and Applications" by Biagini et al.
So the overall answer to the question "Can stochastic integration be further generalized?" is certainly yes.
As regards the question "Is the idea of stochastic integration to accept convergence towards the stochastic integrals in probability instead of almost surely (pathwise)?", there's probably not a simple yes or no answer to this. This is a theory which was developed piecewise with various objectives in mind. To some degree, it is probably fair to say that the answer is yes. Once you note that Riemann sums for the integral $\int_0^t H_s dB_s$ do not converge almost surely even in simple cases, it is natural to ask whether a sensible integral can be defined if a weaker notion of convergence is applied. However, other concerns are present as well. For example, the Ito stochastic integral takes care to ensure that stochastic integrals with respect to local martingales also are local martingales. Meanwhile, the Fisk-Stratonovich stochastic integral takes care to ensure that the conventional chain rule holds. Overall, it is most fair to say that the main issue confronted by stochastic integration theory is that various types of conventional regularity of integrals do not present themselves naturally for pathwisely defined integrals, and much theory is devoted to investigate how this issue can be alleviated.
Finally, regarding your question "Can the range of integrators be expanded even further beyond semimartingales if we ask only for convergence in distribution instead of convergence in probability?", this really depends on what you expect of your integral. A way of making this concrete is to ask the question "do the Riemann sum converge more often for non-semimartingales when only requiring weak convergence instead of convergence in probability?" I don't really know the answer to this question. An issue with using weak convergence is that it's a notion of convergence which only depends on distributions, in the sense that the convergence of $(X_n)$ to $X$ only depends on the marginal distributions of $X_1,X_2,\ldots$ and $X$, and not on the joint distributions $(X_1,X),(X_2,X),\ldots$ (in contrast to convergence in probability which depends on $X_1 - X, X_2 - X, \ldots$, and thus uses the joint distributions). This is an issue because of the following. Assume that you are in the case where e.g.
$$
\sum_{k=1}^n H_{t_{k-1}}(X_{t_k} - X_{t_{k-1}}) \to Y,
$$
where the convergence is weak, and $0 \le t_0 < \cdots t_n = t$. Thus, $Y$ is a candidate for the "weak" stochastic integral of $H$ with respect to $X$. Here, $X$ might be some non-semimartingale process, such that the above could be an interesting extension. Now assume that $Y$ happens to be, say, Gaussian with mean zero. In this case, $Y$ and $-Y$ has the same distribution, and thus we would also have
$$
\sum_{k=1}^n H_{t_{k-1}}(X_{t_k} - X_{t_{k-1}}) \to -Y,
$$
where the convergence again is weak. Thus, both $Y$ and $-Y$ are candidates to be the "weak" stochastic integral. This is likely to be an unfortunate property of any notion of stochastic integral based on weak convergence of Riemann sums. Therefore, it's probably not likely to yield a fruitful theory of stochastic integration in this rough form, some modifications or a different idea is required.
