To extract data from DWG file, you need run AutoCAD (locally, or remotely with Autodesk's cloud services) with your custom AutoCAD plugin (or whatever plugin developed by third parties. The data extracting process DOES NOT run inside your web application. That is, your web app is just a server side operation that spawns the DWG data extracting process and collects the extracted data for the users.
It is very possible that you would use Autodesk's cloud service for the data extracting, or you can run AutoCAD's core console EXE (a no-UI AutoCAD on your server, if you host your web app inside your intranet). Either way, you need to be able to develop a data extracting plugin for AutoCAD (or find someone who can do it for you, so that you can focus on the web app for the operation workflow).
It sounds to me that you are not very familiar to AutoCAD .NET plugin development. To get the data as your showed from a drawing file, it would be quite long/steep learning curve, and you need to have decent AutoCAD using/AutoCAD programming knowledge. I'd say, the development of your project should have a 2-person team: someone who does AutoCAD plugin programming very well (and may or may not know web app development well) and the other (maybe, you) who knows web app development well (so it would not be difficult to take advantage of/integrate Autodesk's cloud services into the web app), who does not need to know AutoCAD plugin development much.