Are socket options inherited across accept() from the listening socket?

No, they're not necessarily inherited. Try this sample, which sets the receive buffer size (SO_RCVBUF) on the initial socket to a non-default value and then compares the result with the inherited socket. Run this code, which listens on TCP port 12345, and then connect to it from any other program.

#include <errno.h>
#include <netinet/in.h>
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <sys/socket.h>

void die(const char *f)
{
  printf("%s: %s\n", f, strerror(errno));
  exit(1);
}

int main(void)
{
  int s = socket(AF_INET, SOCK_STREAM, 0);
  if(s < 0)
    die("socket");

  int rcvbuf;
  socklen_t optlen = sizeof(rcvbuf);
  if(getsockopt(s, SOL_SOCKET, SO_RCVBUF, &rcvbuf, &optlen) < 0)
    die("getsockopt (1)");
  printf("initial rcvbuf: %d\n", rcvbuf);
  rcvbuf *= 2;
  if(setsockopt(s, SOL_SOCKET, SO_RCVBUF, &rcvbuf, sizeof(rcvbuf)) < 0)
    die("setsockopt");
  printf("set rcvbuf to %d\n", rcvbuf);

  struct sockaddr_in sin;
  memset(&sin, 0, sizeof(sin));
  sin.sin_family = AF_INET;
  sin.sin_port = htons(12345);
  sin.sin_addr.s_addr = INADDR_ANY;
  if(bind(s, (struct sockaddr *)&sin, sizeof(sin)) < 0)
    die("bind");

  if(listen(s, 10) < 0)
    die("listen");

  struct sockaddr_in client_addr;
  socklen_t addr_len = sizeof(client_addr);
  int s2 = accept(s, (struct sockaddr *)&client_addr, &addr_len);
  if(s2 < 0)
    die("accept");
  printf("accepted connection\n");
  optlen = sizeof(rcvbuf);
  if(getsockopt(s2, SOL_SOCKET, SO_RCVBUF, &rcvbuf, &optlen) < 0)
    die("getsockopt (2)");

  printf("new rcvbuf: %d\n", rcvbuf);

  return 0;
}

Result on a machine running Linux 3.0.0-21-generic:

initial rcvbuf: 87380
set rcvbuf to 174760
accepted connection
new rcvbuf: 262142

Several of the socket options are handled at lower levels of the system. While most of the socket options could be set using the setsockopt. Reference:man setsockopt And since you are mentioning only POSIX on any Linux, in general, as your scope. The accept() (Reference: man accept) does have a certain amount of discretion on what socket options should be inherited and what options to reject from the listening fd.

accept() does not modify the original socket passed to it as argument. The new socket returned by accept() does not inherit file status flags such as O_NONBLOCK,O_ASYNC from the listening socket.

So, instead of relying on the inheritance or non-inheritance of the listening socket properties(which is bound to vary across implementations and licenses), the accepted socket should be explicitly set with the desired socket options.(Best practice)

man pages and the implementation codes in your machine would be the most relevant specification for the accept() behavior.There's no common or standard specification existing across multiple variants of Linux.

Socket options is the place where things go that don't fit elsewhere. So, it's expected for different socket options to have different inheriting behaviour. Whether to inherit or not a socket option is decided on a case by case basis.