Is it appropriate to lift weights at 14?

Start in high school. Around high school age, kids have more maturity (better long-term planning and dedication) and more body image awareness. They also have more testosterone, so their muscles will grow stronger with exercise. Mild or moderate weights are fine, but heavy weights can be bad for their bone's growth plates. After they finish their height growth (at age 16-18 years), they can start using heavier weights.
Yes. A well-designed weight training program can be safe and effective at any age. At younger ages, high-intensity maximization of weights is not recommended but a low-resistance high-repetition program can be beneficial. Recent literature has failed to show any adverse effects of a safe weight-training program.