Feynman Diagram and standard model rookie here
In drawing Feynman diagrams one has to be within a model, for the standard model there exists conservation laws that cannot be violated in the Feynman diagrams. Lepton flavor conservation gives a nu_mu coming from conserving muon lepton number. So the correct diagram for muon decay to an electron to first order is
Any loop corrections have to obey lepton flavor conservation.
What you have drawn is a forbidden decay within the standard model .
Extending the standard model to include the experimental observation that neutrinos have mass, will allow the diagram in the question due to oscillations but the specific extension of the SM has to be decided before a meaning can be given to the diagram.
In the Standard Model, leptonic family numbers (LF numbers) would be preserved if neutrinos were massless. Since neutrino oscillations have been observed, neutrinos do have a tiny nonzero mass and conservation laws for LF numbers are therefore only approximate. This means the conservation laws are violated, although because of the smallness of the neutrino mass they still hold to a very large degree for interactions containing charged leptons.
However, the (total) lepton number conservation law must still hold (under the Standard Model). Thus, it is possible to see rare muon decays such as µ → eγ or µN→eN:
Searches for these decays have come up negative, just giving limits. Feynman diagrams for neutrino oscillation effects can be drawn .
Is it even possible that a particle will stay around waiting for another particle which will get created later, so that they may interact? Wouldn't the neutrino speed away before the anti-neutrino is created?
Feynman diagrams are iconal representations for planning the mathematical intergrals necessary to get a value for the decay rate ( in this case). Within the diagram the "particles" are virtual, i.e. not on mass shell because they will be integrated over the limits of the problem given. It has no meaning to think of "waiting", everything happens within the limits of integration.