I've seen Fiverr used to build fans to a new social network page or profile, build links (spammy) for a website, or design a logo. Is it an effective tool for building fans to a social media page (in terms of, for example, creating an illusion of popularity) or are the fans acquired worthless?
Generally no. Bot accounts do not interact with your posts on Facebook, meaning your page will get zero social activity and will look dead to any normal person.
It's much better to email your personal network of people asking them to help you out and like the Facebook page. This will give you a much healthier base of fans to work with and build upon.
It depends. You most probably won't get real fans for $5. Usually those are bot accounts.
But having 500 bot-likes can be helpful. Real people tend to "like" stuff that has some likes already.
The only risk is that when you over-do this facebook might decide to close down your account.
I agree with all the above. For five dollars you can set up a Facebook ad and gain real fans from the FB community. This gives you the social interaction you are looking for.