After how many seconds are file system write buffers typically flushed?

Let me restate this your problem in only slightly-uncharitable terms: You're trying to control the behavior of a physical device which its driver in the operating system cannot control. What you're trying to do seems impossible, if what you want is an actual guarantee, rather than a pretty good guess. If all you want is a pretty good guess, fine, but beware of this and document accordingly.

You might be able to solve this with the right device driver. The SCSI protocol, for example, has a Force Unit Access (FUA) bit in its READ and WRITE commands that instructs the device to bypass any internal cache. Even if the data were originally written buffered, reading unbuffered should be able to verify that it was actually there.

There are a lot of caches involved in giving users a responsive system.

There is cpu cache, kernel/filesystem memory cache, disk drive memory cache, etc. What you are asking is how long does it take to flush all the caches?

Or, another way to look at it is, what happens if the disk drive goes bad? All the flushing is not going to guarantee a successful read or write operation.

Disk drives do go bad eventually. The solution you are looking for is how can you have a redundant cpu/disk drive system such that the system survives a component failure and still keeps working.

You could improve the likelihood that system will keep working with aid of hardware such as RAID arrays and other high availability configurations.

As far software solution goes, I think the answer is, trust the OS to do the optimal thing. Most of them flush buffers out routinely.

The only way to reliably make sure that data has been synced is to use the OS specific syncing mechanism, and as per PostgreSQL's Reliability Docs.

When the operating system sends a write request to the storage hardware, there is little it can do to make sure the data has arrived at a truly non-volatile storage area. Rather, it is the administrator's responsibility to make certain that all storage components ensure data integrity.

So no, there are no truly portable solutions, but it is possible (but hard) to write portable wrappers and deploy a reliable solution.

First of all thanks for the information that hard disks lie about flushing data, that was new to me.

Now to your problem: you want to be sure that all data that you write has been written to the disk (lowest level). You are saying that there are two parts which need to be controlled: the time when the OS writes to the hard drive and the time when the hard drive writes to the disk.

Your only solution is to use a fuzzy logic timer to estimate when the data will be written.

In my opinion this is the wrong way. You have control about when the OS is writing to the hard drive, so use the possibility and control it! Then only the lying hard drive is your problem. This problem can't be solved reliably. I think, you should tell the user/admin that he must take care when choosing the right hard drive. Of course it might be a good idea to implement the additional timer you proposed.
I believe, it's up to you to start a row of tests with different hard drives and Brad Fitzgerald's tool to get a good estimation of when hard drives will have written all data. But of course - if the hard drive wants to lie, you can never be sure that the data really has been written to the disk.