Oh boy, I could probably write quite an essay for you on roleplaying githyanki - the best and longest PBP game I've ever played (and am playing) is essentially about exploring githyanki culture, mindset, and history.
However, I'll try to be brief and practical minded.

First, I would keep in mind that the giths of the Laughing Gallows are pretty non-classical - they are more inspired by the pirate giths from Spelljammer and assumed to have little if anything to do with those derived from the traditional Astral empire. Still, if you're playing a character that diverts from the cultural norm, its a good idea to know a bit about what the norm is like.
So to get a short summary on how the Planewalker community suggests you play typical githyanki, for instance, follow
this link and turn to page 23.
Also, I would highlight this paragraph from the same section:
Personality: Cold, aloof and utterly militaristic, the githyanki have little in common with the distantly related humans. Although exceptions are quite notable, most githyanki only focus on war and domination – all life is defined by these. Needless to say, competition runs deep in the githyanki psyche. Even everyday activities become silent contests to outdo each other. They have a need for people to be organized hierarchically, like a military unit. Surprisingly, though, violence almost never erupts from this competitive drive. Ever since the separation from the githzerai, githyanki never fought other githyanki – as long as they were loyal to their lich-queen. Even with Vlaakith dead, the githyanki remain a proud and loyal race.
So I wonder, how would a Gith think? I don't know much, if anything, about their culture so I took the racial features and broke them down into some base assumptions. These assumptions I tried to also fit into the idea of the culture of the Isle of the Hanging Gallows.
1. Danger Sense: +2 Init. What exactly is a danger sense? Being as the Gith get +2 to Int also I thought that these two features might be slightly related. Its not that they're quick to recognize danger but perhaps rather a paranoid analysis of those they meet. They're always suspicious and as such, are always expecting something could go wrong.
2. +2 History. Okay, this threw me off. Githyanki may have a natural affinity for picking up the lore of places they've been to. They have a desire to travel and as such pick up things from the places they go to and of places they may wish to travel to. As such, they often compare things people say to what they know of the places those people are from. I'm not sure how to include this though really.
3. Gith Willpower: Stubbornness is a trait in and of itself. A refusal to ever give up or to let someone else push you down. A brain that can deconstruct the littlest conversation will never give in to attempts for control.
I would not recommend basing your roleplaying approach heavily on racial abilities, but instead read what lore you can find - "A Guide to the Astral Plane" is a great source, but there's also some in the various monster manuals.
Still, I'll try to adress your thoughts on the racial abilities in 4E.
1. Githyanki are extraordinarily vigilant and pretty paranoid. You pretty much nailed that.
2. Githyanki have immense appreciation of their racial history, I guess the 4E writers figured that extends to history in a broader sense as well. Every single githyanki that has been raised within their tradition has a very detailed knowledge of their history (even if those teachings are very one-sided). All of them know broad strokes such as:
- They were enslaved in the ancient past by the illithids.
- They were used and manipulated as tools both mentally and physically.
- They rebelled and brought down their oppressors under the leadership of Gith.
- A group of their brethren disagreed philosophically and split off to become the githzerais (in the eyes of githyanki a fundamental betrayal that directly has lead to the mind flayers surviving).
- Gith "sacrificed" herself to forge the dragon pact (specifics are unknown, but conspiracy theories also suggest she was betrayed by Vlaakith).
etc...
3. This seems very explicitly an evolutionary thing. They could likely not have succeeded in their rebellion if they had not over time grown resistant to the mental domination of their illithid masters.