"Can smaller sensors have more pixels per inch?"
yes, but number of pixels doesn't tell you nearly as much about image quality as sensor size.
"Bigger sensors capture more light, but again, you can make up with that with more sensitivity, up to a point."
Yes, sensor quality and sensitivity can make up for sensor size to a point. You can have a crappy camera with a big sensor, but there are optical limits that cannot be overcome without a larger sensor.
"Are the optics easier to design with a smaller sensor?"
smaller sensor = smaller optics = cheaper to manufacture
"If the lens is bigger, you have a smaller depth of field, what about the sensor size, must affect the optical properties."
The smaller the sensor, the smaller the optics. Also, the same optics with a smaller sensor will result in a larger zoom. The small sensor will only pick up some part of the image acquired by a larger sensor, so the smaller sensors image will be more zoomed in. This is why p&s cameras report there optics as 35mm equivalent. The actual optical lengths change with the sensor size, so they convert them to 35mm equivalent so that people can meaningfully compare them.