So can I do automation without speakers and headphones?

If you don't want to mix [which includes automation] on proper studio monitor speakers for whatever reason, then you don't have to -- you can do all your work in GB while listening on the built-in laptop speakers, or earbuds, or whatever you want. I recommended working on a proper pair of speakers, but everything you do can be done just listening to the sound from the laptop if that's what you prefer. And that's really all I have to say on this topic.

I can read that mixing engineer does sound mixing, sound design, game audio, audio editing, audio engineering. From audio I will only use vocal. Do I need all of them then? There are written mixing and sound mixing. Are they different things?

Those are not aspects of a single project, they are different types of projects. Different kinds of mix engineers work on different kinds of projects -- if your recordings are music projects with vocals that that's what you'll be doing. I have no idea what "written mixing" is.. ?

"Written mixing" means that I wrote that there was an advertisement, where mixing engineer offers two of his jobs - mixing and sound mixing. What is the difference between "just" mixing and sound mixing.. ?

? All mixing is sound mixing -- I have no idea what that person meant, maybe it made sense in context of the rest of what he was saying..?

I enclose a video, you can see, mixing, sound mixing and other jobs they are offering. Do I need them all if I have a song with vocal? And mixing is what ever it could be by them, but sound mixing is the mixing we are talking about? https://youtu.be/0xMD1F8EMmc

I don't know what distinction they're implying with those two different mixing categories -- you'll have to ask them. Whether or not you need for-hire mixing services depends on what your projects are like.. Do you have a finished song with a lot of tracks that need to be balanced and processed? If so, then maybe you might benefit from these guys' services. If you don't have a finished song yet, or if you only have a vocal track, then you don't really have anything that's ready to mix, and it's premature to be worrying about mixing. I don't have any idea what your music projects are like, so I can't advise you on what you might need in that regard.

I checked out again the video and realized maybe mixing is a category (you see a sign near it) and other labels are jobs what will be done under mixing category. Right? OK. I don't have any song yet, thus I do not worry about mixing anymore.

Will all sorts of audio processing be done only by plugins? Firstly I use plugins/audio processing by myself and then hire a mixing job about used plugins?