It's probably not possible to really do it. You could play something like Swat 4 while role-playing an one of the Special Circumstances agents I guess? You can switch between the views of each member of your team, use a fiber optic device to look under doors, sometimes have snipers set up, and look through their scopes/take shots.

Lots of what we see "on screen" (but obviously it's a book) is really a somewhat detail-less alien action book dressed up as 1990s Earth (that's why the characters are called humans, and why certain things are described inaccurately).

A good entry in an action game series started in the 90s isn't a bad bet. As I said, it's close to what good parts of the books portray themselves as anyway.

Play SimCity or Civilization and contemplate what your relationship is to each of the sims or population units digitally represented in the game is. That would be the difference in level of consciousness between the Minds and ordinary humans like you and I. You are making those crude representations of beings in game happy, protecting them from outside threats, and generally looking out for their well being but your ability to relate to them or converse with them as equals with them will never be any different.

Can’t think of one. Maybe something like Commander Blood - high concept space opera, focus on alien interactions, humourous & quirky. But it’s also nothing like it.

If you like puzzles, look at The Talos Principle 2. It's mainly about asking whether humanity should even try to control the physical universe and spread life, or focus on sustainability and safety. The Culture itself referenced a couple of times in it.

Interesting question. Stellaris is the clear choice if you're into grand strategy, you can be an AI-led multi-cultural civilization (starting or become one) with Culture-like win conditions. Endless Space II is another option that's turn-based and a little more approachable.

Mass Effect has a lot of fun with a galactic society. If you like the crew-based action of Consider Phlebas and want more of a story then that or (believe it or not) the Guardians of the Galaxy game that came out a couple of years ago are good choices.

Deus Ex and Cyberpunk 2077 tackles trans-humanism to an extent but they're Earth-bound. Primordia is a point-and-click adventure where humans are extinct and our creations live on. I could probably think of some more but the dinner alarm is going off!