How many transmissions does it normally take for a neuron to reach threshold for action potential?
How many action potential are needed to cause an action potential in the postsynaptic neuron depends on the strength of the synapse(s) involved. It's not possible to answer this question in general.

It is not just the number of action potentials received but also the firing pattern in which they arrive and the location of the synapse(s) on the dendrites/soma (proximal/distal, on a spine or not, on which kind of spine,...) of the postsynaptic neuron.

However, for some neurons it is possible to trigger an action potential in a postsynaptic neuron with a single action potential. But in this case it is very probable that multiple synapses were involved between the two neurons (example paper).

To your second question: the time until polarization wears off is called the time constant tau of a neuron. It is a property that varies from neuron to neuron and is one of the things that can be measured when recording from a neuron. It is one of the many factors that affects how incoming action potentials are translated into a neuron's output firing.