There are several minutes of explicit material that got cut, I'd say around 10 minutes in total, but trying to get a precise count would be a bit tedious. There is a sex scene with one of the dancers early on, a meeting in the shower with another girl, a scene with the radio host, some nudity with the dancers outside of the sex scenes, and I think some other minor stuff I am forgetting.

Hey thanks for even that kind of rough estimate and detail!

It says that preorders end on 3th July.

Does it mean that 3th July is when the game will ship or that is still unknown?

I never bought from Limited Run Games, but If my memory serves me right, they basically never ship right away (expect 3-4 months after the order on average), although they take the money beforehand.
You are welcome! If you are interested in the original footage, you can see a tinyurl link in one of the posts above, that's where I got to download the originals, so I could compare them with what's available in the game. I am not 100% sure if the "PG 13" version linked there is exactly the same used in the game or not, but that's also available to download.

The item's page would tell you any ETA if they had one. The standard ETA is 8 - 9 months after pre-orders close. Some exceptions happen from time to time (like the base version of TMNT Shredder's Revenge), but the item's actual webpage would tell you if there was a shipping estimate or not.

Can one still order the game after the preorders are over? Or are those limited runs...limited?

Limited Run Games' business model is on FOMO so once the pre-order is gone, you won't be able to get it from Limited Run anymore.

Thankfully there are other publishers out there who produce physical games of indies and sometimes they push out the same games as what previously came from limited run. Takes a bit of research and googling but the best think to do is contact a dev via twitter/social media and find out if they're printing physical and with which partner.

And, after a while, the unrated version found it's way to GOG...
https://www.gog.com/en/game/american_hero_unrated