If Abraham was 'as good as dead' and unable to procreate, how could he father a child through Hagar in Genesis 16?

There are a couple of assumptions to expose. The verse you want is not Hebrews 11:11, which in the preferred reading actually begins with Sarah, not Abraham; the verse is Romans 4:19. And the translations that render it 'as good as dead' are not quite correct; the Greek word is katenoesen, from kata and noeo, meaning to think deeply or consider. So the text says Abraham thought, or considered, that his body had died; it does not say he knew for a fact he could not reproduce. Any normal ninety-nine-year-old man would think that, and Abraham perhaps had what today we would call a temporary condition that made him doubt. But he was still able, and Abraham could not know human biology; God did, and God called him at the right time. In fact, from the genealogy in Genesis 11 you can see that the patriarchs' primary childbearing years averaged about age thirty-two, close to our own, and every man retains the ability to father a child until he dies; so ninety-nine is out of the ordinary but not impossible. That answers the timing between Genesis 16 and 18: Abraham may have thought he could not procreate, but he still could. Sarah at ninety is the harder case; she was well past the childbearing years, so God may have had to intervene miraculously in her case, and that is something I would want to study further.