Just a note: I'm a beginner myself and I'm not that experienced in the view of emptiness and self but I'll try to help you with the concept of anatta from my limited understanding.
This article by His Holiness The Dalai Lama might be of interest: Emptiness and Existence. I'll try to answer your questions by deferring to the Dalai Lama's own words because he explains the concepts of self better than I could ever. Just a note: this is the view held by Tibetan Buddhists. You'll get vastly different views on the self and what passes during rebirth from other sects.
I won't answer your question about what's passed on in rebirth because that's a subject I'm woefully ignorant on. I'm pretty sure that most Tibetan Buddhists hold the view that the subtlest mind, the level of mind that only exists at death, is what's passed on because it doesn't exist dependent on the coarse physical body or nervous system. But I don't completely understand it myself.
If there is no self...
We have to address this first. From my limited understanding, there is a self that exists conventionally by being merely designated by us. Ultimately, there is no inherently-existing self, a permanent essence that defines you. There's no inherently existent soul, self, essence, whatever you want to call the concept of a permanently existing thing that defines a person. That doesn't mean you don't exist, that just means you don't exist independently from all phenomena. You exist but dependently on other causes and factors.
The Dalai Lama talked about it in "Approaching the Buddhist Path" Chapter 2 (The Buddhist View of Life) Section 2 (Body, Mind, Rebirth, and Self):
“Generally speaking, once a mind conjoins with a fertilized egg, a new life has begun. In dependence on that body and mind, we designate “person,” “living being,” “I,” or “self.” Our outlook on life, perceptions, feelings, and emotions are all based on the notion of a self. We say, “I did this. I think that. I feel sad or happy.” Although this is our experience, seldom have we stopped to ask ourselves, “Who is this I upon which everything is predicated?” The question regarding the identity of the self is important because it is the self, the I, that wants to be happy and to avoid suffering. If the self existed independently from other phenomena, we should be able to isolate and identify it.
The Buddha taught that a person is composed of five psychophysical aggregates — form, feelings, discrimination, miscellaneous factors, and consciousness. The form aggregate is our body, and the other four aggregates constitute our mind. If we search among these five aggregates, we cannot pinpoint a person who is totally separate from them; nor can we identify a person who is identical to his or her body and mind. The collection of the two also is not a person. A person exists in dependence on his mind and body, but is neither totally one with them nor completely separate from them.
The self depends on the body: when our body is ill, we say, “I’m sick.” If the self were a separate entity from the body, we could not say this. The self also depends on the mind. When the mind is happy, we say, “I’m happy.” If the the self were a separate entity from the body, we could not say this. The self also depends on the mind. When the mind is happy, we say, “I’m happy.” If the self were separate from the mind, we couldn’t say this. On the basis of the mind seeing a flower, we say that the person possessing that mind sees the flower. Other than this, we cannot find a person who sees something.
In short, “I” is designated in dependence on our body and mind, yet when we search for a findable thing that is “me,” we can’t find it within the body, the mind, the collection of the two, or separate from them. This indicates that the person exists dependently; we lack an inherent, findable, unchanging essence. Since we lack an independent self, we can change, grow, and progress from confusion to awakening.
The person or self creates the causes for happiness and suffering. The person also experiences the pleasurable and painful results of these actions. Although we cannot pinpoint anything that is the self, the existence of a person who creates causes and experiences effects is undeniable.
Countless sentient beings have this feeling of self, although it is difficult to identify what that self is. However, the fact that each and every sentient being wants happiness and not suffering is indisputable; no reason is needed to prove this. Being born, enjoying life, enduring suffering, and dying are conditioned phenomena that are products of previous causes. If there were no person who experiences them, none of these would be tenable. Similarly, we distinguish afflictive saṃsāric existence from the awakened state of nirvāṇa based on whether the person has gained realizations. This distinction between saṃsāra and nirvāṇa would be irrelevant if no person existed.
The self exists, but from the perspective of the deeper nature of reality, our view of it is mistaken. This incongruity between how the self actually exists and how we apprehend it is the source of all our confusion and suffering. As we cultivate correct views, our mental strength increases; this leads to mental peace, which, in turn, brings joy and a sense of fulfillment. A mental state that sees the world as it is is free from fear and anxiety.”
That's all I can answer. Hopefully someone more knowledgeable comes by and also answers the other questions.