Is time going backwards beyond the event horizon of a black hole?

It's easy to forget that, in the context of relativity, there is no universal time. You write:

For an outside observer the time seems to stop at the event horizon. My intuition suggests, that if it stops there, then it must go backwards inside. Is this the case?

But your intuition doesn't seem to take into account that, for an observer falling into the hole, time doesn't stop at the event horizon.

The point is that one must be much more careful in their thinking about time within the framework of general relativity where time is a coordinate and coordinates are arbitrary.

In fact, within the event horizon, the radial coordinate becomes time-like and the time coordinate becomes space-like. This simply means that, to "back up" inside the event horizon is as impossible as moving backwards in time outside the event horizon.

In other words, the essential reason it is impossible to avoid the singularity once within the horizon is precisely that one must move forward through time which, due to the extreme curvature within the horizon, means moving towards the central singularity.

Your intuition is misguided, time does not run backwards inside a black hole. For an observer inside a black hole, time passes in a perfectly "normal" way, such as it does at the horizon. The stopping of time of time at the horizon is, as you mentioned, a phenomenon that only an outside observer experiences. It can for example be measured by noticing change of the received frequency of light signals which are emitted from near the horizon at constant frequency.