First, let me say that there are vast amounts of criticism toward the book Sex at Dawn by experts on the topic of the evolution of human sexuality, but generally for reasons other than homosexuality. See here. But that's not what you're really asking about.
I believe there is possibly some miscommunication going on here. Some of the statements in the question, I think, are true and makes sense. Some statements are not totally clear, and the answer depends on how exactly they are interpreted. Others are wrong. Let's see.
In order to do this, she then goes on to show how, in the animal world, the great majority of all sexual activity is directed only towards reproduction.
It is possible that this is true statistically. I'm not sure. However, there is certainly many examples of non-reproductive sexual behavior. See the Wikipedia article: Non-reproductive sexual behavior in animals. One obvious example is the Bonobo.
Another author, David Barash, says similar things: namely, that "evolution is very much about reproduction—variation in reproductive success is evolution".
This statement I think is far more reasonable. It is true, in the ultimate sense, that the main purpose of sex is reproduction. And behavior that increases the probability of reproduction is going to be selected for. (Technically, it would be more true to say that natural/sexual selection is about variation in reproductive success. Evolution contains more than selection, for example drift).
However, homosexual behavior can easily arise naturally in the animal kingdom, especially as a side-effect. One way this can happen is the following: feeling pleasure through sexual stimulation is adaptive, because this increases the probability that individuals engage in sexual behavior, which ultimately leads to more offspring. However, as a side-effect of sex being pleasureable, this means that animals may engage in sexual behavior that is non-reproductive (e.g. homosexuality or masturbation).
As long as non-reproductive sexual behavior is not done to the exclusion of reproductive sexual behavior, there is no reason to suspect that the non-reproductive sexual behavior negatively affects reproductive success (and hence, non-reproductive sexual behavior can perfectly well exist).
Despite the original ultimate reason for sexual behavior is to facilitate procreation, sexual behavior may even possibly become adaptive for reasons other than procreation. For example, it may start facilitating social bonding.
This naturally leads to the following problem, when looking at the following quote:
"Properly speaking, homosexuality does not exist among animals…. For reasons of survival, the reproductive instinct among animals is always directed towards an individual of the opposite sex. Therefore, an animal can never be homosexual as such."
When asking whether homosexuality is common in nature, we need to define precisely what we mean by homosexuality. If a human man says that they're homosexual, that can mean several things. It may be that they are attracted to mostly men, or both men and women, or only men. If they are attracted to only men, they are, what I would call, strictly homosexual. If given free choice they would in a majority of situations prefer to have sex with another man, I would call them primarily homosexual. In this sense, our Bonobo male friends are typically neither strictly homosexual or even primarily homosexual, because a male that engages in homosexual behavior is also willing to engage in heterosexual behavior. Strict homosexuality or primary homosexuality does appear to be quite rare in nature (although there are some examples known. For example, in rams a few percent seem to prefer same-sex even when given the choice). In the quote, when they say 'properly speaking' they may be talking about what I have denoted strict homosexuality. The quote is a bit hard to interpret. If they simply mean homosexual behavior, then no, that is not particularly rare.
Is this true? Is homosexuality really uncommon and unnatural among our closest relatives who are primates?
Strict or primary homosexuality is uncommon, homosexual behavior is fairly common (and therefore clearly not extremely unnatural).
Is sex mostly for reproduction?
In the ultimate sense, the original and most important purpose of sex is reproduction. But as I've detailed out, non-reproductive sex can still easily arise without it negatively affecting fitness. It may even possibly become adaptive (if it is not done to the exclusion of reproductive sexual behavior).