Uji monogatari (2nd part after Genji dies) is basically all rape and a pretty intense criticism of it to boot, but let's go back to Genji since that part is often left out in the English versions (and it is possibly written by someone else given the sudden shift in tone).

Some of the women are raped, most are also in love with Genji, and all of them are in complicated court power dynamics and courtship regulations at the time. A few of them are complicated courtships, in that they are not really given the chance to refuse and it is taken as an unfortunate reality of court life. Arguably the most intentionally damaging actions are from the mad ghost of Rokujo towards the women Genji sleeps with (she outright murders Yuugao) rather than from Genji himself. We also have one instance where Genji is refused and has his way with the lady's brother, but we don't know how the brother felt about it.

However, keep in mind that Genji Monogatari is a work of ficfion from a very court-critical lady-in-waiting to a very exclusive group of educated women. Unfortunately modern readers tend to read historical fiction with a tendency to see it high class and "how things should be done" (see how people miss Shakespeare's toilet humour). A lot of the book was written to entertain and even crticise court life.

1. Ladies of the court were very, very bored and love affairs were considered a pastime activity. So the complicated courtship (and hearing about forbidden love) was a means of entertainment. However, the lady listeners were also subject to similar advances from other court men, so this could have also been cathartic (as they did not necessarily enjoy the visits themselves, so might as well listen to a story where the undesired visitor is a young and handsome figure of power).

2. Murasaki repeatedly wrote in her diaries how drunken and stupid the men of the court were, and much of genji monogatari can be seen as reflecting that. She detested court life and its social activities. Irony and sarcasm were a thing, and it is possible that many of the love affairs in the book were written with that intention.

Additionally, This is more of a theory and not proven, but some writers believe Murasaki Shikibu was not into men. Aside from her arranged marriage (only two years - she never remarried) and woes about her patron courting her (implications given power dynamics at the time do unfortunately point to her being raped, though she never mentions it; only that he never won her love but still pursued her, and if is known that his patronage was necessary to make a living). So it is possible that it is just how she sees men courting, and without explicitly calling it rape and devasating (aside from uji monogatari) she is clearly showing unhappy situations for genji's women.

Oh, my God, I'm so glad you're asking this question. The answer is mostly “it is complicated and people have different opinions.”

First of all, we may ask if Heian Japan had a definition of rape. It did. As introduced in the Ritsuryō codes of 757, there were two possibilities for illegal sexual conduct; gōkan 強姦, forced sexual misconduct, and wakan 和姦, mutually agreed sexual misconduct. Both of these referred to a married woman having sex with someone other than a husband either forcefully or willingly. So we do have a concept of rape legally speaking, HOWEVER, this is massively complicated by the fact that this was a law framework that was basically taken wholesale from Chinese law and did not match very well onto the Japanese concept of marriage, which at the time was extremely nebulous and not as clearly defined through patrilineal lines as the Chinese one. It is also unclear how much these were applied to actual marital interactions at the time. For this reason, we tend to not use these concepts when we talk about relations in Heian tales, since these are never really brought up. We absolutely see expressions of sexual assault being decried in some Heian narrative texts, but it is not necessarily along sexual lines. In Konjaku monogatari shu, for instance, a man is considered immoral for taking a woman’s clothes after raping her, and negative descriptions of these sort of interaction only really occur when the man perpetrating it is of lower rank than the woman (Something that in most cases would not apply to our Shining Prince). So, these concepts are not widely used at the time, but they do exist.

Now, let’s go back to Genji. Feminist critique of Genji has traditionally divided itself along the lines of “Genjigirai,” Genji haters, those who believe Murasaki Shikibu wrote the tale particularly in order to illustrate the position of disempowerment of women at court, and those who believe Murasaki’s opinion on her hero is a positive one. Scholars like Royall Tyler, one of the most respected translators of the tale to English are firmly on the side that the tale very much describes an elegant game of seduction where the performance of vulnerability is key for women to attract men. A more nuance take on this is subscribed to by Margaret Childs, who used to believe that performance of vulnerability was an important tool to gain support from men, but now acknowledges that the performance of this vulnerability came with an actual vulnerability that needed to be negotiated and that these women were often in a position of physical impotence and inability to stop the sexual act from being performed.

Though not by any means perfect, an interesting take is that of Otilia Milutin who looks at descriptions of physical and literary reactions to sexual relations in the tale, as well as the reactions of the present ladies-in-waiting to determine the instances where this vulnerability is being “performed” and where it is as genuine response. For instance, though tears are to be expected, if the lady in question is also sweating with embarrassment and fear of what people may think she sees this as not performative, since it is an expression of uncontrolled physical reaction.

The most interesting tool she offers is the presence or not of the morning-after poem. This poem is a Heian staple and absolutely necessary after a night of sex. Often this is the poem that will consolidate the night of love into a long-term relationship or establish it as a one time thing and also what the lovers will use to reassure each other about their feelings about the events. Milutin points out that the lack of a morning-poem response can be seen as a clear indication of the lack of consent in Genji, such as for instance when Genji has sex with Murasaki for the first time (also pay attention to the description of the sweat).

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He spent his idle hours with her, just playing go or character games, and the young woman displayed such charming and endearing traits, even while they played their silly games, that, although for months and years he had felt her childish charm without thinking of anything more, he now found it difficult to hide his feelings any longer. She will find it painful, but what was he to do? So, even though there appeared to be nothing particularly different in their relationship, one morning came when he woke up early, while the young woman did not wake up at all. Her ladies-in-waiting were all worried, “What could be happening? She is not behaving as usual.” Just as he was about to depart, Genji pushed a writing box inside her curtains and then left. While no one was around, the young woman innocently raised her head and found beneath her pillow a knotted letter. Innocently, she opened it and read:

[poem] “Why did you and I spend night upon night separated, our garments alone embracing each other.”

 He wrote it in a casual hand. She had never even dreamed that he might be considering such things and she wondered dejectedly how in the world could she have trusted with all her heart someone so horrible.Around noon, he returned. “You seem to be ill. How are you feeling? It’s no fun if we don’t play go today.” He peered in inside her curtains and there she was, lying prostrate, with her garments covering her head. As he approached her, her ladies-in-waiting all drew back. “Why must you pout so? You are, unexpectedly, such an unpleasant person. Think of how odd your ladies will find this behavior!” He pulled off her bedclothes and found her bathed in perspiration. Even the hair on her forehead was soaking wet. “Well no, that’s not good. This is really improper behavior,” he said, trying this and that to improve her mood, but she really resented him and refused to give him an answer. “Well then, you will not see me again. I feel very much hated,” he complained and opened her writing box to peer inside. There was nothing in it.

That said, it is clear that Genji does include instances of consensual relations, such as Genji and Yugao, where it is the lady that makes the overtures to the relationship. Generally speaking, we can say it is complicated, but also there is clearly a notion of both impotence, inability to act and acknowledgement of the lack of agency of women when it came to sexual assault, both within and outside the tale.

Sources:

Yoshie Akiko. Family, Marriage and the Law in Classical Japan—an Analysis of Gender in the Ritsuryō Codes on Residence Units" (Presented at USC, 2013) (This one can be found online for free, for those interested!)

Tonomura Hitomi. "Black Hair and Red Trousers: Gendering the Flesh in Medieval Japan." 1994

Childs, Margaret "The Value of Vulnerability: Sexual Coercion and the Nature of Love in Japanese Court Literature." 1999

Childs, Margaret "Coervice Courtship Strategies and Gendered Goals in Classical Japanese Literature" 2010

Milutin, Otilia. Sweat, Tears, and Nightmares: Textual Representations of Sexual Violence in Heian and Kamakura monogatari. 2015

I am curious about Part III and Ukifune. It seems to be a criticism of the treatment of women in the court, but it's also taking place during a time when a "great light" has gone out. I assume that light is Genji. But are some of the gods gone too? Is the world revealed to be imperfect in this part of the novel, whereas it was a better world in the earlier half when Genji was alive?