A scalar times a vector is a vector.
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It helps to understand division as the opposite of multiplication. In this case, v / s = x; a vector divided by a scalar is something unknown. Turn this around, into a multiplication: x times s = v. In other words: What must I multiply by a scalar to get a vector?
Scalar
The product of two vectors can be done in two different ways. The result of one way is another vector. The result of the other way is a scalar ... that's why that method is called the "scalar product". The way it's done is (magnitude of one vector) times (magnitude of the other vector) times (cosine of the angle between them).
Time is scalar
No it is not a vector