see

What Do Bioinformaticians Use To Make High Quality Web Sites?

How To Learn To Build A Bioinformatics Oriented Web Site

for some direct answers:

1. Depends on the exact required specs but you'll probably need AWS or GCP or similar for compute if its non-trivial and/or will have many regular users
2. If you are serious about building a website from scratch you will most certainly need to use Javascript at least for the front end. You should be able to find a web server framework amongst C, C#, and Java. You could also look at R Shiny for an all-in-one front end/back end framework, though its usefulness is limited once you move beyond basic R usages.
3. The only language that can run inside a web browser is Javscript. So no, you cannot run Perl "inside" the web site (front end).

The best option though is to look at the numerous other data analysis/workflow management system frameworks that already exist, some of this include web front-ends: https://github.com/common-workflow-language/common-workflow-language/wiki/Existing-Workflow-systems

Notably, all of this has already been implemented in Galaxy, so you will probably want to give that one a thorough review as well: https://github.com/galaxyproject/galaxy

Note that there is a public version of Galaxy available as well, so you wouldn't have to build anything at all if you could wrap your analysis pipeline in a Galaxy workflow and get it uploaded there.