Does each sensory neuron have its own connection the brain?

I'm not a great expert - it's a long time since I studied the nervous system, and I'll star your question in case someone else knows better but no, they don't. There are things called reflex arcs where a nerve dives into the spinal column and then straight out again to operate a muscle. Also, if I remember aright, some neurons synapse onto other nerves before arriving at the brain.