For your first question:
If $X$ has the homotopy type of a CW space, then you can replace $X$ by any CW space $Y$ that is homotopy equivalent to it (in the unbased sense).
Then $Y$ is also $A_\infty$ with a good basepoint. Then you can use the homotopy extension property to make the basepoint a strict unit for the multiplication.
As to your last question:
I think the answer is yes. For example, one can use Boardman and Vogt description of the classifying space to give a description of BX which doesn't require that the unit be strict.