Developer options are a set of tools which may be useful to people doing development, and for those who like to tinker or have deeper control of the system, but aren't needed by most people. That's the reason why the menu is disabled by default (though IMO hiding it via an Easter Egg is rather childish).
Root is something different: that involves giving the user (you) the ability to run any app with administrative privileges, which normally only some pre-installed apps would have. That is useful if you want to have full control over the phone, with the understanding that includes the ability to mess it up in ways that you wouldn't normally be able to. In the early days many people used this to work around the limitations of Android hardware and software (particularly storage, allowing SD storage to be used fully for apps and app data, for example). These days there's much less need, but some people still prefer to have the option to remove unwanted system apps, do more comprehensive backups than are otherwise possible, block ads system-wide or apply more robust firewalls, or even replace the installed OS with a modified (custom) version. Unlike the developer options, which are available on any phone, this isn't officially supported by any major manufacturer. But there are varying degrees of tolerance: some manufacturers make it quite easy, provide an official bootloader unlocking tool, or make source code available to developers, others add security features that make it increasingly awkward to do. In general it's all a bit less controlled than Apple devices.
I'm afraid that I don't really understand your question about third party apps. There are bad apps, but Google have a relatively open policy and will allow useless apps, even those that invade privacy while offering little in return, as long as they don't step over into being outright malware or violate Play Store rules. You just have to think about what it is you are installing.
