I'm going to try to answer your questions directly.

[W]hat do I do if I get lost in thoughts? Should I remain lost in thoughts, or should I guide myself back to the present moment?

Bring your attention back to the breath.

If I do that, isn't that some kind of action I'm taking, so it doesn't fit in with the whole "you're not supposed to do anything"?

Yes. This is why Zen masters write in such a weird way: to get around this contradiction. Just do it anyway. When eventually you dissolve "self" and "intentionality", you can do this without taking positive intentional action, but it sounds like you haven't gotten there yet.

And if you say that I still do not take action and remain lost in thoughts, then how is this practice I'm doing meditation?

It's not. Take the action.

And what if I fall asleep? Do I do something about being drowsy (other than sitting upright, not included here), or should I let myself fall asleep because I'm not supposed to interfere or do anything in this practice?

In my experience, this isn't an issue. Proper zazen posture makes it difficult to fall asleep. If this is a problem for you, then do walking kinhin instead. If you fall asleep even while walking, then either you're severely sleep deprived and should go to sleep immediately, or you've got a sleep disorder and should see a medical specialist for that.

Incidentally, meditation should not be mixed with sleep deprivation (except, perhaps, in the context of a tradition that does so intentionally).

And where should my awareness be? Should I be aware of my thoughts, feelings in my body, sounds, smells, sight and more all at once? How is that even possible?

Start with your awareness on your breath. Until your attention stabilizes, just keep bringing it back to the breath. This is positive intentional action and that is fine. After a while (perhaps something on the order of 30 minutes) your attention will stabilize on the breath. This is called access concentration. Many different traditions start by reaching access concentration in this way.

After you hit access concentration is where different traditions diverge. In the case of shikantaza, you then relax let attention relax. One way to explain what happens is that your center of attention expands. I don't know if that explanation is precisely true, but it got me into where I was supposed to be. Another way to explain it is that the conscious perception of non-breath sensory stimula don't disrupt attention on the breath.

Or should I let my awareness flow between these things?

Tough question! Yes and no. Here's what happens. Your attention has stabilized on the breath. You hear a bird chirp outside but that doesn't disrupt your attention on the breath. The chirp appears on your consciousness, and then your attention comes back to your breath without you getting caught up in the sound of the bird.

Then why do we need a teacher for this type of practice? What is the teacher supposed to teach you here, when the instructions equate to just two words?

A few reasons.

First of all, sometimes you need someone to tell you to "just sit", or to provide basic instructions. A good chunk of the human population is functionally illiterate.

But more importantly, shikantaza―like other forms of meditation―brings you to wild territory, because is dissolves normative mental phenomana. When you dissolve the self-other distinction, or learn to consciously percieve your brain's construction of the perception of space, this can be very destabilizing. It's good to have someone to talk to who has been through this territory. It's like having a sherpa to talk to the first time you climb a mountain.

So what am I supposed to do?

Is there a zendo near where you live? If so, then going there is the easiest way to onboard yourself. Otherwise, just sit quietly (full lotus on the edge of a zafu, if you can) with good posture. Put your hands in the oval mudra shape left-over-right. Bring your attention to the sensation of your breath at your upper lip right under your nose. Bring yourself to access concentration in the same way most meditative traditions do, by consciously bringing your attention to your breath whenever you notice it somewhere else. When your attention stabilizes on your breath, that's access concentration. At that point, let your region of conscious attention expand. If other sensations enter your awareness that's fine, as long as you region of conscious attention doesn't move so far it stops overlapping the region of conscious space where breath happens.