How can Hollywood have trademark rights to the image of the Hollywood sign

Another twist -- the other attorney answers are good ones -- is that the word "Hollywood" is also geographical. Those are particularly unusual...

Can I use an existing brand name for a completely different product in a different industry?

If it is a famous mark then you risk a trademark dilution situation. Better to develop your own brand.

Would it be smart to trademark my company name before the company grows too big?

Assuming no one else has already registered (search www.uspto.gov at a minimum, registration gives you valuable rights. For Federal registration...

It seems unlikely you could legally import to the US or resell fakes in the US. I suppose that theoritcally you could 'buy' them and own a bunch...