Simple answer: yes

While your familiar is within 100 feet of you, you can communicate with it telepathically. (From spell description you cited)

Before you go to sleep, you can communicate clearly enough to leave the owl instructions to be alert and then do something if it detects something. (Wake me up, hoot as loudly as possible, etc). The text you quoted states that it will obey your instructions. Since it can act independently of you, you can be asleep and it can be awake.

Your familiar acts independently of you, but it always obeys your commands. (same citation)

• Quite frankly, your points 1 through 5 are an over-complication. If your table / DM goes into that level of detail and verisimilitude, then this particular detail needs to be discussed or resolved with your DM via the DM's ruling on how this works. It's best that you clear this up with your DM before your next play session if you want your owl to keep watch for you.

But there's a catch. The owl is not guaranteed to detect danger.

From the Basic Rules, DM, p. 39 (owl's stat block, excerpted)

Owl
STR 3 (−4) DEX 13 (+1) CON 8 (−1) INT2 (−4) WIS12 (+1) CHA 7 (−2)
Skills Perception +3, Stealth +3
Senses darkvision 120 ft., passive Perception 13

Keen Hearing and Sight. The owl has advantage on Wisdom (Perception) checks that rely on hearing or sight. (Emphasis mine)

While it's likely that the owl will detect danger that can be seen or heard, since it has darkvision and advantage on hearing/sight Wisdom(Perception) checks, a monster or NPC approaching by using Stealth may foil the owl's efforts at detection.

The owl may or may not detect danger, depending on whether or not the stealth check is successful as compared to the owl's perception check. If the stealth is more successful, then having the owl on watch won't result in an alert. This is the same risk as having a PC on watch: they may still get snuck up on if the approaching creature's Stealth score beats the PC's Perception score.

Experience

In our first campaign, my brother's wizard used an owl familiar that he assigned to help keep watch while he was trancing/sleeping while another character (like one of the 3 humans) was on watch. On a few occasions this helped alert us to approaching danger.

See? Yes. Notice? Maybe.

The rules on Hiding and stealth give a lot of leeway to a DM, so you are absolutely within your rights to declare that once a creature is looking directly at you, you cease to be hidden from it. But keep in mind a couple of important distinctions:

1. Hiding is different than being unseen

Any creature that is in a heavily obscured area, or invisible, or fighting a blinded enemy is "unseen". But a creature that is hiding is not only unseen, but also unheard, and generally unperceived. They might keep very still, or stand close to similarly colored parts of their surrounding, or compact their body so their silhouette no longer seems humanoid. In short, they are doing more than being out of line of sight.

As evidence of this distinction, consider the following on page 177 of the PHB:

An invisible creature can’t be seen, so it can always try to hide. Signs of its passage might still be noticed, however, and it still has to stay quiet.

To put it another way (and to quote a 2016 Errata in the PHB):

the question isn’t whether a creature can see you when you’re hiding. The question is whether it can see you clearly.

2. You can't Hide from a creature that sees you, but maybe you could remain hidden

The rules are clear that

You can’t hide from a creature that can see you clearly, (PHB, p. 177)

but note that this only rules out becoming hidden: it doesn't necessarily rule out remaining hidden. So as a DM, it will totally be up to you whether or not the Goblin will notice the already hidden rogue when they gain line of sight.

Keep in mind that combat is a chaotic and distracting place. As an example of what that level of chaos can do to perception, note how difficult it is to count how many passes the players in white make in this linked video.

Note, though, that once the hidden rogue attacks anything, they won't be able to hide (from the goblin) again, unless they've moved to some new location that the goblin can't see. The rules are clear that while you could remain hidden while visible, you can't hide.

3. Advantage/Disadvantage (on attacks) is not from being hidden, it is from being unseen

All the points above are meant to indicate that a creature could remain hidden while they are seen: that is, an enemy might not notice them, or might not know where they are. But that does not necessarily mean they will gain advantage on an attack roll against such an enemy.

Any place that the rules suggest a hidden creature will gain advantage on an attack, they justify this as a result of the hidden creature being unseen. Such as:

Master might allow you to stay hidden as you approach a creature that is distracted, allowing you to gain advantage on an attack before you are seen. (PHB, p. 177, bold added)

And:

When you take the Hide action, you make a Dexterity (Stealth) check in an attempt to hide, following the rules in chapter 7 for hiding. If you succeed, you gain certain benefits, as described in the “Unseen Attackers and Targets” section later in this chapter. (PHB, p. 192, bold added)

So the only real benefit a hidden character could get against an enemy that can see it is that the enemy might not notice them. This could cause an enemy to be surprised (first round of combat only) or to be unable to intentionally attack the hidden creature (because they don't know it's there). But a creature hidden in plain sight would not get advantage on attack rolls against a creature that could see them, regardless of whether or not they remain "hidden".