Is performing ghusl in the following manner valid: taking water from the tap with one's hand and pouring it over the head and body, then washing the feet?

Doubt concerning the emission of semen does not necessitate ghusl (ritual bath). The default state is not being in a state of janabah (major ritual impurity) until certainty is achieved in that regard, and performing ghusl at that point is not obligatory. What suffices for ghusl from janabah is to make water reach all outward parts of the body, along with running water through the hair so it reaches its roots. The fulfillment of this, or a strong presumption of it, is sufficient.