I realised recently that my shorts have a netting like swimming shorts do and decided to ditch the boxers and see how it was. What an absolute game changer! I don't know why I didn't do this a year ago when I started running.
I exclusively run in 2-in-1 shorts that have built in spandex like liners. Some are more 'compression' than others. Most are just supportive, not compression. I like the Nike 2in1s and they last a LOT of miles. I've tried a few cheap amazon brands and not liked them nearly as much. The liners are never as supportive.
I only buy unlined shorts, and wear my own boxer briefs. The shorts last way longer than the liner does, in my experience, so it's easier to just separate the two functions into two separate items. It's the Unix philosophy.
Always underwear. Aside from the netting being coarse on my delicate skin down there other people don't want to see my cock swinging around so blatantly in my shorts. I've seen it on other people and it isn't a good look. Also swinging free like that is super uncomfortable for me, I want my shit locked down and not slamming into my legs!
It’s a personal preference. For me, I only buy shorts without liners and wear my own from Saxx. I tried a bunch of different built in options about 15 years ago and ran into uncomfortable compression or chaffing issues.