It is, albeit a bit buried in the formal specification.

When I trying after remove all connectors, NuGet.exe install command works without any exceptions. :(

This error means that ASP.NET is not installed/enabled; it's a setting that's available in the server manager console, role/feature settings. Here is a guide from Microsoft on how to enable this: http://www.iis.net/learn/get-started/whats-new-in-iis-8/iis-80-using-aspnet-35-and-aspnet-45

You can look in the Web.config file in the ProGet Service directory to see what connection string it is currently using to connect to the database. If you installed everything with default settings, this will be using integrated authentication, so it's connecting using whatever account the ProGet Service is running as. Outside of this, whoever installed/updated ProGet last would have also needed to have access for integrated auth.

I just installed v 3.4.6 Build 6 and it is now working. I used SQL Express installer this time instead of connecting to a stand alone sql server. Not sure if that made any difference.

Thanks for the report - there is a new version available from the Extension Gallery (v4.1) that should resolve this.

To install ProGet, you must be an administrator and have db_owner privileges against the target database.