What Can Heimdall See?
Almost everything. From the descriptions, it sounds like he generally sees the big picture, unless he deliberately chooses to look for something specific.
In-Universe Descriptions:
From here I can see nine realms and ten trillion souls.
- Heimdall, Thor: The Dark World
It’s said the gatekeeper can see a single drop of dew fall from a blade of grass a thousand worlds away.
- Volstagg, deleted scene, Thor
There are secret paths between the worlds to which even you, with all your gifts, are blind.
- Loki, Thor
You think you can deceive me? I, who watch all? I, who can sense the flapping of a butterfly's wings a thousand worlds away? Or can hear a cricket passing gas in Niffelheim?
- Heimdall, Thor
The second instance in which Heimdall's vision fails [the first being his inability to see the Frost Giants who entered Asgard]:
Heimdall: I turned my gaze upon you in Jotunheim, but could neither see nor hear you. You were shrouded from me, like the Frost Giants who entered this Realm.
Loki: Perhaps your senses have weakened after your many years of service.
Heimdall: Or perhaps someone has found a way to hide that which he does not wish me to see.
Out-of-Universe Descriptions:
MCU:
Superhuman Senses: Heimdall possesses sensory capabilities far beyond that of any Asgardians, which made him the gatekeeper to Asgard. His sight itself extends itself to all of the Nine Realms, and his hearing is so precise that he heard the Warriors Three and Sif conspiring against Loki before he decided to passively help them. Heimdall was able to hear Thor call to him from Earth. Despite his phenomenal ability, however, his senses can be deceived by Loki and the Dark Elves.
- MCU Wiki
Heightened Senses: Heimdall is gifted with sensory capabilities far beyond those of other Asgardians. He is able to tune his awareness to see and hear events as they occur in far off realms as if he were standing in the near vicinity of those he observes, (Omni-Senses)
- Marvel Comics Wiki for Heimdall (Earth-19999)
Comics:
His senses are said to be so acute that he could hear sap running through trees and see events occurring miles away.
- Marvel Comics Wiki
Heimdall possesses extraordinarily acute superhuman senses, most notably his vision and hearing. His senses are said to be so acute that he could hear sap running through trees, and "the tiniest plant growing in the heart of the hidden hills", and see and hear anything occurring in Asgard or on Earth. Heimdall can also "look across time, as well as space", in one instance seeing the far-off approach of an invading party and correctly predicting that they were still a full two days away from Asgard; this ability to see what has yet to come is retained even after the establishment of the new Asgard on Earth.
Heimdall has been said to be capable of detecting the fluttering of a butterfly's wings "a thousand worlds away". Heimdall is able to sense the life essences of Asgardian gods throughout the Nine Worlds of Asgard, and has the ability to focus on certain sensory information or block it out of his consciousness as he chooses, being so alert that he requires no sleep at all. According to Ares in his War Plan for the Siege of Asgard, Heimdall can "see and hear as far as creation", even capable of seeing the Dark Avengers at Avengers Tower from Broxton.
- Wikipedia
What are Heimdall's Limitations?
This question is best answered by considering the cases in which he fails to see something:
In the first two cases, Loki was deliberately obscuring his vision [see the exchange between Loki and Heimdall quoted above]. In the third case, Thor was obscuring his vision:
After the funeral of Frigga, Odin decreed that the Bifrost be shut down. Not needing to guard his post, Heimdall went to console Thor. As they spoke, Thor asked Heimdall to side with him as he hid Jane Foster, who was carrying the Aether. When Odin was told by Tyr that the mortal woman was missing, Odin looked disappointingly at Heimdall.
- Marvel Cinematic Universe Wiki
It appears that, in the MCU, Heimdall's vision has only failed in situations where a god is interfering with it.
Wikipedia suggests that there are other limitations:
Certain magical spells can block his sensory powers. Additionally, his powers of sight are limited if even one of his eyes cannot see; apparently his eyes are capable of sustaining physical injury and require a long time to naturally heal, if ever. At least once, Heimdall has shown the ability to project an avatar (as an enlarged image of his face) to others from just outside Broxton to Manhattan, as he did with Thor, though he claims that he cannot maintain it for long, as his power has diminished with the fall of Asgard.
- Wikipedia
Hydra and WWII:
This is difficult to answer. There are any number of possibilities:
Heimdall could have seen it, but didn't know that he should be looking for it.
He couldn't see it, but we don't know why.
He did see it, but it was never mentioned in the movies - and why would it be mentioned? Captain America was about Captain America, not Heimdall or Thor. Since the premise of Thor was that Thor and the Asgardians hadn't been to Earth in some time, it would be strange if it was revealed that Asgardians took part in World War II. Heimdall's voiceover in Thor makes it clear that Asgardians and humans have somewhat strained relations:
Questions, they've always asked
questions -- this race called man,
on this planet they call earth.
Passionately longing to know how
they are connected to the heavens
**In ages past, they looked to us as
gods, for indeed so many times we
saved them from calamity**. We tried
to show them how their world was
but one of the Nine Realms of the
Cosmos, linked to all others by the
branches of Yggdrasil...
...the Worlds Tree. Nine Realms in
a universe of wonder, beauty, and
terror that they barely
comprehended.
**But for all their thirst for
knowledge, they let our lessons
fall into myth and dreams**.
Where did he come from? He came
from us, the proudest race of
warriors the Worlds have ever seen.
He came from this -- the greatest
Realm the universe has ever known.
He came from Asgard!
This quote strongly suggests that Asgardians have, in the recent centuries, taken a "hands off" approach to Earth.
