I have a non-immigrant B visa and a work permit. The visa expires ("enter before" date) in two weeks, but I just entered the country, so the stamp ("admitted until") is valid for 90 days.

 

Is it correct that I/my company should apply for an extension of stay 30 days before the stamp date ("admitted until"), even though the visa will already have expired at that point? And that means I cannot leave the country until that process is finished, right?

After you enter the country the visas validity means nothing.

You will apply for the extension before the end of the 90 day entry from the visa. You can do it at anytime during the last 30 days of the 90 days.

If you want to leave the country during the 90 day entry you apply for a re-entry permit to keep the remainder of it valid when you enter the country.