UEFI can boot from SATA and NVMe disks
As with any other firmware type, UEFI can boot from devices that it has drivers for. Newer UEFI systems will generally have drivers for NVMe; older ones may be SATA-only.
UEFI can boot from MBR and GPT disks
Yes, although the OS being booted may have its own restrictions. It's usually said that Windows doesn't like UEFI-booting from MBR disks.
(Note that booting from an MBR disk doesn't actually use the MBR boot code; only the MBR partition table.)
EFI partitions are just FAT32 partitions labeled EF00 or such code. Unclear about this
Both MBR and GPT have a "partition type/purpose" field for each partition. In MBR, the type is 0xEF for the EFI partition (vs 0x07 for a Windows data partition).
In GPT, the type for an EFI partition is C12A7328-F81F-11D2-BA4B-00A0C93EC93B, which the gdisk partitioning tool abbreviates to EF00 for convenience.
FAT32 is the mandatory baseline that all UEFI implementations must support.
UEFI boot does not read the MBR and looks for "boot entries" that must be manually (or installed by the OS) configured
It doesn't read the MBR boot code, yes. It looks for "boot entries" that are stored in the motherboard's NVRAM (the "CMOS memory").
Boot entries point to a specific file within a specific partition, e.g. \EFI\Ubuntu\grub.efi.
For removable disks (and often for internal disks also, if no boot entries are present), UEFI will also look for a bootable .EFI file at a fixed path within each EFI-compatible partition it finds.
What I don't know and it is my main doubt: does UEFI MUST have a EFI partition to boot?
Technically, no. The boot files must be somewhere on the disk, but it doesn't necessarily need to be a dedicated partition.
For removable disks (USB sticks) a dedicated partition isn't required at all; often the .efi bootloader file is just on the main partition of the USB stick.
For fixed disks, it's usually needed because the standard type of an EFI partition is FAT32 and you don't want that for the main OS partition. However, since UEFI implementations may support other filesystems, I have actually seen systems that have the EFI files dumped on the main Windows NTFS partition – the user's firmware supported NTFS and just used whatever partition the boot-entry told it to use.
(The "partition type" ID isn't actually checked when booting; boot entries specify the partition's unique ID instead.)
That being said, you really should have it as a separate partition, and especially if your OS is going to use something like Ext4 you won't be able to pull the "single partition" trick because your firmware does not support Ext4.
On the same context: how does this EFI partition (AKA ESP) relates to the so called secure boot? What exactly secure boot means?
Secure Boot means digital signature checks on everything that's booted. Mainly, this means that the .efi files that you'll have in the EFI partition have digital signatures embedded in them (the same format as .exe files) and the firmware will enforce that.
Even if I somehow manage to convert the disk to GPT and UEFI I'm not sure how I will configure the dual boot (currently GRUB takes care of that) but that's theme for another question
If you don't use Secure Boot, you can continue using GRUB for that (and it will usually autoconfigure to recognize Windows as before). Launching one .efi app from another is a thing.
The cleaner method would be to have two UEFI boot entries, each directly going into the corresponding OS bootloader (e.g. you select "Ubuntu" and get GRUB). The entries will show up in the same boot menu that you get by pressing F8 or such.
