Yes to both questions, with some caveats.
A single avocado tree will fruit. However, there are two different flower types. If you plant an avocado tree of "flower type A" near a tree that makes "flower type B" they will both bear more heavily. But from what I understand, a solo tree should make enough for home use.
As to the second question, it takes avocado trees 5-7 years to start bearing from seed, and occasionally a seed-grown tree will take much longer. Avocado varieties also cross readily so the fruit size and quality are unknowns until it starts to fruit. Also, it can take a couple of months for a pit to send out roots. (On the other hand, I have also seen pits that started putting out roots inside the fruit.)
Last comment: there are several cultivars with enough cold tolerance for your area. But some cultivars are extremely frost-tender. If you buy a tree, you should check its cold tolerance. If you grow seed from a locally-grown avocado, it should be tough enough for your area.
Sounds like fun, however you do it.