I've heard that Linux has several advantages over Windows, like it loads files faster and manages memory better.

So my question is, will a Linux OS like Debian or Ubuntu perform significantly better than Win7 or Win10 in terms of overall performance when hosting a single server using Dream Daemon, on the same hardware?
Yes. Just by nature of how many resources Windows itself with little running uses, you'll definitely end up with more resources to play with on a headless Linux box. Mind you, if you use a GUI you'll lose some of those advantages.
if it is just a hosting machine i like CentOS(no gui) nice small distro but enough to do the job.
I highly recommend either CentOS or Ubuntu LTS.
Ubuntu LTS or even Ubuntu Minimal is the way to go for me, minimal edition runs idle at like 20 MB RAM. The only issue is that its stripped down a lot and you just need to install the necessities.