Do I understand correctly that there is one unique and unique Linux kernel, on the basis of which all these numerous distributions are developed (Ubuntu, Debian, Mint, etc.).
Not really. Only one base (initial source code). That is, distributive developers download the source codes of the Linux version they need , make some corrections (patches) into them, set build parameters to include the necessary ones and exclude unnecessary features, and compile the kernel. In other words, distributions differ not only in repositories and application programs, but also in kernel variations.
... and at the same time it is possible to install this purest kernel, and not one of the distributions?
Can. But since the kernel is distributed in source codes, you will have to compile from them your own version, sharpened specifically for your computer and, possibly, needs - the regular configurator make menuconfig is rich in options:

The main window of the graphical configurator for Linux build parameters. Source: Wikipedia
However, obtaining even a minimal operating system is a rather lengthy and multi-step process that requires sufficient knowledge of the principles of Linux operation.
Do I understand correctly that the kernel itself is essentially an OS, only without a graphical interface (something like DOS) and the GUI interface can already be installed separately?
And again, not really. The kernel is just an interlayer between the programs and the hardware. In the case of Linux, there are still drivers included.
To get something similar to DOS, you will additionally need at least GNU Coreutils - a set of programs that define the user environment ( ls , cat , su , etc.), and the bootloader (for example, grub , lilo ) that will load kernel when you turn on the computer.
By the way, precisely because of the symbiosis of the Linux kernel and the GNU Coreutils software package, the operating system is called GNU / Linux.
I heard many times that the kernel at the beginning usually needs to be compiled and then installed. Is it possible to get an already compiled kernel, is it distributed in this form?
Distribution in the source code is a kind of policy of the developers of the kernel, and therefore the compiled kernel can only be torn out from some existing distribution.
Is it a good idea to try to compile this very Linux kernel on Windows, and to the extent possible?
To build Linux, you need a GCC compiler and an LD linker. Theoretically , when building under Cygwin, there should be no problems (if it is to compile the kernel), but there is information about the difficulties with case-sensitive file names that Windows considers to be the same.
I heard many times about all sorts of package managers, with the help of which different software is installed on Linux-based systems, and each such OS (Ubuntu, Debian, Mint, etc.) has its own package managers. And what about the clean core? How is something put on him?
Compiled from source. That is, in the case of a clean kernel, you should always have a compiler at hand.
How is Linux software generally distributed? Only in open source, with subsequent compilation? And such things as installation files, .exe-shniki, etc. - are not usually used?
Building from sources is mandatory only for kernel components, such as drivers. Application programs can also be installed in the form of binary files. Go to the developer site, download any package. Then, knowing that any package is just an archive of one of the common formats, unpack it and scatter the files contained inside to their proper places in the system. Finally, make sure you have all the libraries you need to run this program.
If there is only one PC without an installed OS, how is the Linux kernel installed on it?
To build Linux, you need another copy of Linux. Therefore, you should at least get a live cd of some distribution kit.