HACS is the Home Assistant Community Store, a nice way to add integrations from the community to Home Assistant that are not core integrations. HACS allows you to add things without much fuss, and get notified of updates.
You do not need it. You can also add these integrations manually. But if you do use quite a few community integrations, as most of us do, you would probably not want to do without HACS.
But then again, if you think hassio (currently named HAOS) is a complication, then HACS is maybe also a complication? HAOS + HACS is for almost all of us the opposite of complication. But if you want to be in full control and do everything yourself, then you should probably pick manual install for all integrations instead of using HACS to do it for you.
No, HAOS is a complete (stripped) Operating System (OS) with Home Assistant installed inside it (using containers, but also hidden inside). If too is a convenience so users do not have to worry about updating and maintaining the OS, making sure all required software is installed and updated, … It is a complete maintenance free system. Most users use it nowadays.
You can run it from a docker install, but it is installed inside the docker using a script. See the link I posted if you push the button Start using hacs, which takes you to the page below, where you choose the tab named “container” (which is what you have):