What happens to an astronaut (with a long rope trailing behind him), when he crosses the event horizon of a black hole?

Dale's answer is correct, but I want to further emphasize that nothing special happens in the vicinity of an event horizon. It's just like any other region of spacetime.

Here's an analogy. Suppose you're in a building that's rigged to explode at a certain time. If you're in the building and too far from an exit at a late enough time, you won't be able to escape before the explosion even at your top speed. If it's a single-story, square building and you can exit at any point on the edge, then the region from which you won't be able to escape at a given time is square. It starts in the center of the building and expands outward at your maximum running speed. The boundary of that region is the "escape horizon".

If you don't escape and die in the explosion, then the escape horizon will necessarily sweep over you at some point before your death. When it passes you, nothing special happens. You don't notice it passing. You can't detect it in any way. It isn't really there. It's just an abstract concept that we defined based on our foreknowledge of the future.

The event horizon of a black hole is defined in the same way, with a singularity in place of the explosion and the speed of light in place of your running speed. If your worldline ends at the singularity, then the event horizon will sweep over you, at the speed of light, at some earlier time. But you won't notice. You can't detect it in any way. It isn't really there.

People get confused about this because there's phenomenology associated with black hole horizons: the closer you get to them, the faster you have to accelerate to avoid falling through, the slower your clock runs, the hotter you get from the Hawking radiation, and so on. They also behave like electrical conductors for some purposes, though it's not mentioned as often.

The thing is, if you mispredict where the singularity is going to be, and try to escape from what you think is the horizon but actually isn't, all of those same things happen. Any event horizon defined by any future spacetime points, whether singular or not, has these properties, even in special relativity. (See Rindler coordinates and Unruh effect for more about the special-relativistic case.)

So the answer to any question about what you'd see while falling through an event horizon is always the same as if the event horizon was somewhere else.

He would still see the rope trailing behind him. There is nothing that prevents light from the rope to fall from the rope inward through the event horizon and to his eyes.

It depends on your point of view. For the astronaut, nothing, in particular, happens.
For us, as distant observers, the astronaut will get frozen in time. Someone once wrote (in this question I read on this site) that upon entering the BH's event horizon, the astronaut's hand will be pulled off his body, due to the fact that time stands still on the horizon. This is obvious nonsense. For the astronaut, nothing seems to be going on. He's just falling freely through the horizon. Looking back he just sees the rope falling with him. And because the BH's mass is that big he won't get spaghettified (this may seem strange because of the enormous mass of a galaxy-like BH, but the radius of the event horizon is huge too).
If he/she is hovering above the event horizon ( which takes an enormous amount of energy) put's his/her into the hole, he/she won't see, feel, or is able to pull his/her hand back. When they start moving away from their stationary hovering position, they...I'll let you think about this.

His/her information content is completely lost and we won't be able to tell from the Hawking radiation what he/she looked like. Contrary to the ADS/CFT correspondence approach which is based on string theory and states that information isn't lost. I don't accept string theory to correspond to reality, so neither do I accept that information isn't lost.
We will never know what will happen if she falls on and on. Maybe she'll end up in another Universe, maybe she'll get spaghettified, but for sure she's lost forever.

In short: Because someone (in a special suit) is in free fall and as long the someone is not torn apart by tidal forces, that someone will observe nothing strange. He/she would see everything as here on earth. It's only for us as distant observers that time- and one space-coordinate take each other's place. In free fall, the time, t-, and spatial (radius, in the case of polar or spherical coordinates), r-coordinates don't interchange.