Is it possible to have arthritis in your knee after getting surgery?

Yes. Arthritis is wear and tear in the joint....I often say "rust in the joint." this can be from many things - years of physical activity, being overweight, genetics, rheumatological problems...To name a few. Even a complete knee replacement can have degenerative changes around the replacement (although less so). Staying active and flexible are some of the best ways of helping this.
Sure. Really depends on nature of surgery but short answer is yes. If the surgery was to "clean up" a torn meniscus this can lead to arthritic change in the knee long term. The less shock absorber you have the greater stress the cartilage will see and the more rapid the wear.