For the past two years I've been running MX Linux with Xfce. For now, it seems to offer the nicest x86 distribution without compiling your own. Up to date. Easy install.
I have a EeePC 1000HA 2GB RAM 120GB SSD - They're getting pretty old and limited by needing a 32 bit OS. I mostly use it for a ssh terminal and as long as it works I'll keep it.
Yes, you could try them all without having to install them (using something like Rufus to "burn" the ISO to a USB stick). I think Peppermint would be the best choice in your case, though. If it's too slow, then try the others. Bear in mind that they will all run slower from USB than if they were installed.
Nothing complicated really. Any OS that runs on this hardware is fine since you only need to check if it's a hardware issue or your currently installed OS is broken. Running an OS installer will be enough.
I can suggest Lubuntu (32-bit version), for example - it runs perfectly on this device and has a pretty lightweight and simple installer. I'm using Balena Etcher for making bootable USB sticks.
Yes, you can use a USB wifi dongle, something like this should work. I use a Bluetooth version with mine, as it doesn't come with it built-in.