Being "completely free of responsibility" is difficult. You can minimize your risks but there are always risks. People can sue for any reason they can imagine. It may not stand up in court and a trivial complaints might be handled by a lawyer with a single billable hour, but you'll still have the expense of fighting off the complaint. You'll want to talk with a lawyer about the things you need for your own project. Even putting up a WordPress site can run afoul of disclosure laws for cookies, from logging web addresses without a ToS, from not having a published data privacy policy, from recording any information from minors without a COPPA policy. They're all quick to set up, but there are unscrupulous people who search for them and try to extract money from violators. Business lawyers have many variations to choose from, so they're worth the visit.
Preventing people from removing banner ads is impossible. Their computer displays it, so unless you somehow have complete control over their computer (which you don't) then a skilled developer can remove it, even if that means the simple process of turning off their network connection while playing. Very few people will take technical steps to remove ads, and smart developers will provide an in-game transaction to remove ads; if you know you would likely make under $1 per account from ads, adding a $1 or $2 purchase to remove ads will both give the players a better experience and give more money to your pocket. It is one of the most common IAP options.
Assuming you're in the US, protecting logos and visual designs are part of trademarks. Protecting the product from redistribution are part of copyrights. You should discuss both with a lawyer. Copyright registration is quite cheap, $35 if you do it yourself and read the fairly simple registration instructions. Trademarks cost $225 and have moderately complex rules that make it worthwhile to have a lawyer do it. Either way, the protections mean nothing unless you can actually enforce them, which means the cost of going to court. Even a very cheap lawsuit where you can get a settlement can approach a six-figure sum.
As for names, keep clear of names that others have already used. Search on the various search engines for the term, search the federal trademark database, search your own state's trademark database, and avoid anything that sounds like, looks like, uses part of another name, or otherwise brings in any recollection of any other existing product.
