Can I upload my web server as a docker to Lambda and have it run forever there? I though Lambdas were supposed to be more short lived (like a couple hours), is that not the case? It's been a while since I actually looked at Lambda because GCP run is so clean.
If you're married to running your own Web server, you could use schedules to run your Lambda every 15 mins....but you might as well use Fargate at that point (easier and probably cheaper).
If you're not married to running your own Web server, you can use API gateway with your Lambda functions, which is the traditional approach. Your frontend and it's assets can be served from an S3 bucket with Lambdas powering your backend. Cold start times will be a concern with this approach, but there are "warm-up" strategies you can employ to prevent that.