I think it is not possible to give a clear "yes or no" answer to your question, because it is a question about a research area where there remain many models which do not agree with one another, and we simply don't know which if any are right. The research area being inflation theory.
Inflation or something like it may have happened, or it may not have happened. The biggest unknowns here are to do with entropy. Attempts to model the early universe in detail typically evoke (without always realising that they have done so) extremely special states of affairs. This makes it hard to assess whether or not a given theory has not so much explained something as shown that it would be the outcome of something even more inexplicable. Inflation does not escape this problem.
I think the main message here is that something rather odd is happening in our day in the interaction between research science and the wider public. The distinction between carefully constructed and tested ideas and mere speculation is blurred in many popular books, and You Tube channels are even worse. In elementary particle physics, progress over the last 80 years has required a partnership between experiment and theory. There are occasional examples where theoretical understanding put in place something well out of the range of experiment but which proved to be correct (Higgs mechanism being a good example). But there are also many examples of cases where experiments yielded surprises. Inflation is an attempt to grapple with physics at the energy scale $\ge 10^{15}$ GeV. Experiments have accessed up to $10^4$ GeV.
I think the best way to respond to your question is to encourage continuing interest in these areas, but also to encourage a greater role for the attitude "well we really don't know yet".
But one thing we do know is that every scientific model ever put forward for anything has invoked a continuity between one thing and another, between a prior situation and a consequent situation. The idea that physics suggests that something could come from nothing is simply a misdirection, a deliberate miss-use of words, presumably in an effort to gain readers or something like that. I mention this simply because the title of your question suggests that you may have been miss-directed into this sort of juggling with the meanings of words.
Among the authors well-placed to comment here, and who does a reasonably balanced job I think, is Sean Carroll.
Added edit to answer specific point at the end of the question.
Either with or without inflation, space is reckoned to have started from an early state presumably described by quantum gravity, and it grew extremely fast at early times. This resulted in energy density fluctuations being present on pretty much all distance scales. This is modeled theoretically by using quantum theory to provide a value for the standard deviation of the distribution, and then subsequently treating that distribution as a classical field having fluctuations over space and time with the given standard deviation. The move here from quantum to classical is rather glossed-over in the research literature; it is connected to the subtleties involved in the process called symmetry-breaking.
(What is spontaneous symmetry breaking in QUANTUM systems?)
Anyway the main point for your question is that this is not like Hawking radiation. The fluctuations are already reckoned to be classical, or are treated as classical, whether or not there was a subsequent inflation to stretch them out. (I don't work directly in this research area; I got the above information from a book by Hobson, Efstathiou and Lasenby, and from various review and other papers).