What is the ruling on eating alone, or with friends, or with family in mixed-gender restaurants, and what is the difference between the permissibility of that and the mixing that is permissible only due to extreme necessity?

It is permissible to enter restaurants if one avoids forbidden acts and adheres to the regulations of the Shari'ah. The mere presence of forbidden acts does not prohibit entry; rather, it makes it mildly disliked (makrooh tanzeehi), as Al-Qurtubi said, due to the abundance of falsehood and forbidden acts within them.