Neurons with thousands of connections: where are the extra connections coming from?
As you say, a neuron can have thousands of inputs via its thousands of dendrites.  Each of those dendrites can have a synaptic connection to the (axonic) output of a different neuron.  So the neuron can take inputs from thousands of different neurons, not just from one other neuron.

At the other end of the neuron, the output of the axon can form synaptic connections with many distinct neurons; that's what the branching at the end of the axon is for.

Also, it's misleading to say neurons don't "decide to initiate signals".  Neurons in the brain do "decide" when to initiate an output.  The decision is a complicated function of the arrival of pulses at dendrites from different neurons.  Only some combinations of incoming pulses will cause the neuron to fire a pulse on its output axon while other combinations will not.  The specific combinations can change over time; this forms the basis of learning.

So a brain neuron is in the middle of a very complicated network of synaptic connections from many neurons to many other neurons.