Varsovian:
1. Is it even possible to actually "get into the head" of such a character?
No, but the same is true for any non-human character and even many human characters.
Varsovian:
their thought processes and ways of experiencing reality could be vastly different than those of humans.
Could be, but doesn't have to be, if that makes playing them more complicated.
I'm a big fan of just abstracting stuff, though I realize not everyone is. I don't think a tactician character has to be played by a tactician player, anymore than an linguist character has to be played by a linguist player. The tactician character can simply provide abstract mechanical advantages to their allies, like better initiative or freer movement. So, the thought processes and sensorium of an android (or ghost, or wizard, or alien) can simply have different bonuses and penalties to different things to get across their differences from humans (who in a lot of games are simply the "baseline" from which everything else differs).
Varsovian:
I've played an android character once and my approach was to think of her as of an unusual human... but maybe that was overly simplistic?
Simplistic, yes, but probably not overly.
Isaac Asimov's characters (including his androids) always spoke in English (or whatever the story was translated into) and always used normal, understandable terms for things like cars and pens. He acknowledged that he could at least come up with unusual words for things when writing alien characters, but at the end of the day he felt like that would needlessly complicate things for the reader. RPG players are in a similar situation: is it worth going to a lot of work to make a different character seem different, or is it enough to just highlight a couple of key differences?
Varsovian:
2. I wonder how a society with sentient androids might function? The way I see it, somebody would be building them - so, these androids would most probably get programmed with some version of Laws of Robotics, as well as loyalty toward their creators / owners.
One could go that way, though not everyone would find that terribly realistic. The Laws of Robotics are a handy plot point, which Asimov and others have had a lot of fun with, but they don't represent any form of logical "programming" or design, or even come across as all that plausible. In the real world, I doubt anyone would make robots that were impossible to make into soldiers, and I doubt it's possible to make an artificial intelligence that is inherently loyal.
Varsovian:
On the other hand, if we imagine a society where the androids live free, then - who builds them? Would androids be manufacturing more androids? Would they have any reason for that?
They have whatever reason the game itself requires, including none that is ever revealed. Things in fictional situations don't need reasons, particularly if they're cool. They really only matter if they're a key plot point.
In Eberron, the warforged are rather like androids, in that they are artificial and intelligent. In that setting, they were made for a purpose and it's now illegal to make any more. But that means that finding out more are being made could be an interesting adventure concept. If the adventure is about something else, then what's going on with warforged as a race of people might not matter all that much, though the warforged who appears as PCs or NPCs could still be interesting and strange.
My theory these days is that consciousness is unstable, that anything with even a little ability to think must eventually sleep because each instantiation of their consciousness breaks down after a day or so and must be recreated. So, if I made a game with androids, that would be a factor: if they didn't restart and rebuild every day or so, they'd get worse and crazier until they failed permanently. This goes against one of the common tropes of androids, that they never have to sleep. I'd probably also have my androids eat to supply themselves with energy (and be made out of lightweight, inherently fragile materials, so they didn't have to constantly eat sugar) and drink water to help process the food and keep their materials soft and lubricated.
My androids would be a lot like humans. Possibly smarter, but really that's just asking for trouble anyway.