Is it sufficient to conclude by looking the following weekly demand schedule that the good is Perfectly Elastic?
No, you do not have sufficient information to conclude that the demand is perfectly elastic.

The following figures show two ways that demand curves that are not perfectly elastics could have generated the first two data points in your series:





It should be easy to see how to add other demand curves to generate the rest of the data points.

This is actually a common and more general problem: price and quantity are jointly determined by the interaction of supply and demand. Given two data points, it is impossible to determine whether they were caused by a shift in supply or demand or both. There are two solutions to this dilemma:


estimate a simulatenous equation model that accounts for both supply and demand. You can read about simultaneous equation models in a good econometrics textbook or, for example, in Dan McFadden's lecture notes.
work in a context where you know either the supply or the demand curve are fixed. If you know the supply curve didn't move then any change in price or quantity must have come from a shift in demand and you know that all of your data points lie on (and characterise) the supply curve. The converse is true if you that the demand curve did not move.