The global economy depends on growth, but growth can never be
indefinite in a finite world with limited space and resources.
It is only partially correct to say that the global economy depends on growth, because an economy per se does not depend on it. But it certainly isn't wrong to say so, because a large part of the world's economic activity is funneled into investments. Without growth, there would be few investments and new enterprising going on.
Growth has certainly been one of the primary political goals of the 20th century and continues to be so, as the main technique for progressing the human race and bringing order to the global society. In our new century, it has much to do with continuous improvement of quality of life, and lifting millions (billions) of people out of poverty.
Furthermore, since the growth paradigm is so central to the way we order the economy, much of economic activity depends on it. We count on growth to make our pensions grow so we can afford to retire. Society counts on growth when investing in new expensive infrastructure, so that huge projects will pay for themselves over the long term. The financial system depends on growth of businesses in return for lending out their capital. And so forth.
Moreover, growth always tends to decline over the long term due to the
law of diminishing returns, as we choose opportunities which give the
best returns first, leaving those which give lower returns for later.
Diminishing returns is used to refer to a point at which the level of profits or benefits gained is less than the amount of money or energy invested. This means that after a certain point, it is no longer profitable to invest more to extract more value out of a proposition, and it may be necessary to divert resources to other parts of the economy. For instance, in the early days of the oil industry, there were huge wells of oil to drill and a lot of oil was wasted due to early technologies. Since there was plenty of oil, the spill didn't matter so much. As oil became a scarcer resource and drilling became more difficult, new technologies were developed to counter diminishing returns. With modern efficient technology, we are producing more oil today than we ever have before, even though the oil has demonstrated diminishing returns.
Your assumption is correct in the sense that: the closer we come to the production possibilities frontier which is the current technological limit, we will see a decline in the growth rate. However, because growth is compounded, we have really seen an exponential growth of world GDP the whole time.
To answer your question: with current technology, we would undoubtedly run out of finite resources at one point. That doesn't automatically mean that there would be an 'economic collapse' – instead we would see ever more diminishing returns and a decrease in activity within that particular sector, while more profitable opportunities would emerge. Society would have to change and find a new way of living. There wouldn't be a sudden "energy catastrophe", but a slow and gradual decline over many decennia. It is plausible that we would see lower birth rates, not because people would die, but because it becomes more expensive to have many children while maintaining a certain standard of living.
With that said, as technology progresses, we become more apt at extracting resources efficiently. The big challenge of this new century is to become independent from fossil fuels and base our economy on renewable energy or other sources. There are also other potential shortages, as some rare-earth minerals have supply limits with practical implications for our way of life. Hopefully, there will be a replacement or recycling of these, and/or ways to locate and extract them more efficiently.
The challenges humanity face are certainly great, and no-one knows the solutions yet. But the hope is that we will be able to cope with the change via new technologies and/or changes to society. That doesn't mean that an economic collapse is inevitable, but it does mean that economic, and probably societal, reorganisation is inevitable.
Regarding your other questions, I hope someone else can elaborate on them in a more technical sense:
Could the global economy survive indefinitely with zero growth, while borrowing more
and more money to avoid decline?
Under what scenario would the economy be indefinitely sustainable,
without a huge collapse?