Yes, I believe it is possible to completely erase yourself from search engines. After deleting your four main social media accounts (Facebook, Twitter, Myspace, and Google+), use a tool like Account Killer or Just Delete Me to permanantly erase the rest of your online accounts. This should leave your name clean in search engines.
No, absolutely not. Every time you publish something online, you're not only posting it on that specific site. It's getting cached and being recorded right before your eyes. Sometimes, in the case of photos, you can post it up on one site and find that photo that you've since deleted in Google images being hosted by an entirely different website.
Search engines can throw out the data that you have given up until today but the new searches that you do will put you back in the system. Your searches can also be tracked by your IP address and even though your name is out of the system you can be tracked by your IP address.
Even if you don't put yourself on the Internet, it is possible that someone else has put some sort of information about you on the Internet. I don't think a person can completely erase themselves from the Internet because the Internet is involved so much in our daily lives. Even when you die your information will still remain on the Internet.