virtualbox.org

i have HP envy x360 that came with win10 home

AFAiK, it's capable of running vBox from a hardware point of view.

i7 - 1165G7
16.0 GB RAM
64-bit

but MS says i must upgrade to run vBox and also RDP (remote desktop as a remote win10 tool like TeamViewer).

QUESTION: can i run vBox without handing over money to Microsoft?

You would need to upgrade to Windows Pro to run Windows Hyper-V, but you do not need that to run VirtualBox. (In fact, Hyper-V interferes badly with VirtualBox). The PC I am using now runs Windows Home and has no problems running VirtualBox. All the vms I run on it are Windows, including Windows 10 Pro and Windows 7 Pro.

Microsoft does not care whether a machine in a physical PC or a vm. If it is running a Windows OS, that OS needs a license. You can run a Linux OS in the vm no problem, but a Windows OS needs its own license.

In brief, you do not need Windows Pro to run VirtualBox, but you need a separate license for any Windows OS you run in a vm.