There are now a couple different ways you can accomplish this in AWS:
- The newest (arguably coolest!) is to use Cloudfront to run your code using their Lambda@Edge service. You can completely customize your URL path and have portions used as variables like any other REST endpoint. You attach your Lambda fn to "behaviour" endpoints which give you full URL control. Its fairly deep and beyond the scope of your question to explain it all here, but read through the docs at the link provided and you'll likely see lots of stuff you like.
- Another older, more expensive but more documented method is to use AWS's API Gateway as you have eluded to in your question's tags. It has a great front end console and is easy to connect API endpoints to your Lambda backend logic by attaching them to REST methods. The console helps you "variable-ize" your URL with form field data. This service helps you the most with custom domains to trigger from. Setting up custom domains is a snap in API Gateway. Be sure to use AWS's SSL Certificate Manager for free SSL certs on your custom domain too!
How you specifically setup your endpoints depends on which service you choose. Personally, given your desire to serve up different types of content, I would lean towards CloudFront, and define a "behaviour" URL for your dynamic Lambda content. If the URL request does not match one of your defined behaviours, it defaults to the Cloudfront cache/origin to serve your static assets (somefile.txt). Only matches go to your attached Lambda fn with form data. Very slick!
A lot of example Lambda@Edge fn's are available here.
I have used both and have clients on both now. Lambda@Edge is ridiculously faster and less expensive, BUT is less documented, has a steeper learning curve, and console is not nearly as helpful. I would honestly try both to see which fits your situation and experience level best. Both will get the job done. EC2 is most definitely NOT needed (nor desired perhaps). Hope that helps — good luck!