Right-click menus in a web context?

I am working at a place where we develop expert systems for the finance business. Previously we have had native desktop applications, but moved to web applications during the past years. In our latest solution, we had a web application that replicated a native desktop application in its interaction patterns, design, etc. We are now to develop a new product with the stated goal to "move to the future". How should we reason regarding replicating a desktop application vs. moving it completely to the web (using standard web patterns etc.).

If we are to design a system that follows standard web patterns, how to do with the right-click menu? It seems like something that people have different opinions on, some think it's all native and certainly don't belong on the web whilst others like the efficiency with it and wants to keep it.

Since usefulness is equally (or more) important as usability, is it OK to keep the right-click menu on an "all-web" system, or should we find other ways to navigate/select things? Do you have examples of other ways to do this?

The actions in today's right-click menu can be an action for a specific row and/or several rows