Does AWS Lambda provide a way to make a public route/endpoint for my Lambda functions?
Or do I have to build an entire app to handle routing and invoking Lambda functions?
Yes they do!
As of July 9th they have an api endpoint section that allows you to invoke your lambda function from anywhere!
http://aws.amazon.com/about-aws/whats-new/2015/07/invoke-aws-lambda-functions-over-https/.
You can also use API gateway with lambda to design and test your api.