Windows 11 Home or Pro for a first gaming PC?

Windows 11 Home should be fine for that. Pro doesn't give you a special gaming or streaming boost, so I wouldn't pay extra for it unless you know you need one of its features.

Pro is more useful if you want things like hosting a Remote Desktop connection, Hyper-V virtualization, domain joining, or more control over BitLocker. Those are handy in the right setup, but none is a requirement for playing games or doing a casual stream.

If any of those sound useful, compare the price difference. Otherwise, Home is the simpler choice.

Retail versus OEM is a licensing question, not a performance one. A retail license can generally be transferred to another PC, provided it's only used on one at a time. An OEM license is generally tied to the original device. Check the license terms before buying, especially if you expect to replace the motherboard or build another PC later.

There's also Windows 11 IoT Enterprise LTSC, which some people prefer for a leaner setup. I wouldn't pick it as the default for a first home build, though. Check its licensing and availability before assuming it's a normal retail alternative.

Linux is another option if you'd rather try a free, open-source system. For a gaming and streaming build, check compatibility with the particular games, anti-cheat systems, and streaming software you plan to use before committing to it.