Is it normal that I'm quite underweight and I want to be a bodybuilder?

You wouldnt be mocked. Normal gym ppl like to see fat and skinny ppl at the gym because it shows people are trying to better themselves. Theres this one chic at my gym who was quite fat when she started and now shes really fit. Its cool.

And yes you'll gain strength with repetition. Focus a lot on your legs and squats and also situps.