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From (2), can I automatically infer that AC is a bisector?

From the given figure,

when AC is perpendicular to BD, we won't be able to figure out anything about the triangles or the associated angles and we can't calculate what could possibly be the length of BD.

When BC = CD, C is mid-point of BD and AC is the median, still we can't comment on the length of BD in any possible way.

When both the given statements are considered together, we immediately realize that AC is both the angular bisector of A and altitude of the given triangle BAD. So we split BAD into two equally sized triangles BAC and ACD, such that AD = AB = 5. So, BD is obviously the full-hypotenuse of BAD and length is thus sqrt(2)*5.
_________________

From (2), can I automatically infer that AC is a bisector?

From the given figure,

when AC is perpendicular to BD, we won't be able to figure out anything about the triangles or the associated angles and we can't calculate what could possibly be the length of BD.

When BC = CD, C is mid-point of BD and AC is the median, still we can't comment on the length of BD in any possible way.

When both the given statements are considered together, we immediately realize that AC is both the angular bisector of A and altitude of the given triangle BAD. So we split BAD into two equally sized triangles BAC and ACD, such that AD = AB = 5. So, BD is obviously the full-hypotenuse of BAD and length is thus sqrt(2)*5.

Thank you for the clear explanation. It helped a great deal.
_________________

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