Can I grow eyebrow hair faster by putting baby toothpaste on the over-plucked area?

No. While there are many home remedies to increase the growth of eyebrows including castor oil, coconut oil, biotin and onion juice there's limited data to suggest any real strong benefit. At least, though, these remedies are unlikely to harm you. You do need to be careful about overgrooming though.
No. There would be no scientific explanation that I am aware of why baby toothpaste would cause hair to grow faster.
Nope. You may actually irritate your skin, cause a rash, and make matters worse.