Do biological facts determine when a human fetus is considered alive and human?
Life is generally distinguished from non-life by metabolism and growth. As such, a fetus is alive. The reference to "not...until 26 weeks gestation" that you've heard likely refers to viability.* With the most aggressive medical care, this is the approximate age when a fetus may be able to survive outside the womb. 

The term human from a biologic perspective is a species label.** Given that a fetus is genetically indistinguishable (in broad strokes) from a post-natal human, I think it would be hard to argue that it is anything other than human.

Summary: Yes, a human fetus is both alive and human.


*Note that this use of the word viable is standard but deviates somewhat from the etymology of the word.
**I'm ignoring here other ancient species (homo-) which may be considered human but are irrelevant to the question.