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Can we create a NugetPackage which can support both .NET Core (or .NET 5+) and .NET Framework?

I'm just getting started in NuGet packages creation thing and I know how to create a NuGet package for .NET Framework and .NET Core applications but I'm not sure if there any way of creating the NuGet package where it will support both .NET Core as well as .NET Framework. But I've seen the NuGet packages that works with both .NET Core and Framework.
For example, if I open .nupkg file of NewtonSoft.JSON I see the something like this.
Newtonsoft.JSON hierarchy
And yes, currently I'm developing my packages using .NET Framework 4.5, so it supports both. But I want a native support for that package not with the backwards compatibility in my .NET Core applications. Also, I don't want to develop using .NET Standard, because in Newtonsoft.JSON package it looks like they are using the .NET Standard.

.NET Standard is a specification that has multiple implementations including .NET Framework and Core/5/6.
If you want to support Framework 4.5 in addition to core the most up-to-date version you can target is Standard version 1.1.
ref: https://learn.microsoft.com/en-us/dotnet/standard/net-standard?tabs=net-standard-1-1

Build Error after Upgrading microsoft.net.sdk.functions

My project uses .net framework 4.8. After upgrading microsoft.net.sdk.functions from version 1.0.24 to 3.0.1, the build on my local machine works fine, but there are some errors on the build on VSO pipeline:
microsoft.net.sdk.functions\3.0.1\build\microsoft.net.sdk.functions.build.targets(41,5):
It was not possible to find any compatible framework version
The specified framework 'Microsoft.NETCore.App', version '3.0.0' was
not found
microsoft.net.sdk.functions\3.0.1\build\microsoft.net.sdk.functions.build.targets(41,5):
metadata generation failed.
There is no version 3.0.0 for Microsoft.NetCore.App nuget package. Why would the error ask for that version? Also how can I fix the error?

Looking at the package on NuGet, version 3.0.1 is intended for use with .NET Core 3.0 and higher applications, not .NET Framework applications. Version 1.0.38 appears to be the latest version with .NET Framework support (4.6 and higher).

.NET Core 2.0 DLL loaded on server with just .NET 4.6 installed

Imagine Jane is on Ubuntu and uses VSCode with .NET Core 2.0 SDK to create and push a NuGet package targeting netstandard2.0 to a repo.
Imagine I have a .NET 4.6.1 full framework project - its a Windows Service executable.
Since .NET Standard 2.0 is the common spec in the above and I'm using Visual Studio 2017 Update 3 and latest SDKs and stuff, I reference Jane's .nupkg.
I build and XCOPY my /bin/release to a server VM with just the .NET 4.6.1 Framework installed.
Will it run, or do I need to install anything else on the server?

can not upgrade from EF Core 1.1.2 to version 2.0.0

All, I recently started working on a project that uses EF Core 1.1.2. I saw where 2.0 was released because I started getting errors when I tried to add EF core to a project.
I couldn't update the existing project, I created a brand new console project. Set the .Net framework to 4.6.1 and still had no luck. I always get the following message. Has anyone ran into this issue? thanks in advance
Install-Package : Could not install package 'Microsoft.EntityFrameworkCore.SqlServer 2.0.0'. You are trying to
install this package into a project that targets '.NETFramework,Version=v4.6.1', but the package does not contain
any assembly references or content files that are compatible with that framework.

EF Core 2.0 release has been officially announced and the documentation is (partially) updated.
Of course it's not NET Core only. But the actual prerequisites for targeting Full .NET Framework are as follows:
(1) Project targeting .NET Framework 4.6.1 and above.
(2) Visual Studio 2017 with (important!) 15.3.0 update (also just released) installed
Before updating the VS I was getting the same error. After updating the error is gone and EF 2.0 package is successfully installed.

EF Core 1.1.2 had a dependency on .NETStandard 1.3 or .NETFramework 4.5.1. So as you were targeting .NetFramework 4.5.2 everything is hunky dory.
EF Core 2.0 has a dependency on .NETStandard 2.0. This is not compatible with the .NETFramework, everything is far from hunky dory.
According to the source below you can get .NetFramework support with 2.0 tooling, which is in preview:
https://dotnet.myget.org/feed/dotnet-core/package/nuget/NETStandard.Library.NETFramework
The alternative would be to migrate your application so it doesn't use the .NetFramework but only targets .NETStandard libraries.
Sources
https://www.nuget.org/packages/Microsoft.EntityFrameworkCore.SqlServer/2.0.0
https://www.nuget.org/packages/Microsoft.EntityFrameworkCore.SqlServer/1.1.2
https://learn.microsoft.com/en-us/dotnet/standard/net-standard
Good luck.

EntityFramework Core 2.0.0 is not compatible with a Target framework other than Core 2.0.0. You will need to ensure your project is targetting this and install any dependancies.
It will NOT work with the (standard) .NET framework versions, it is a 'Core' specific release.

Can I use "Roslyn" in Visual Studio 2017 with .NET 4.7?

I've upgraded my project with scripting capabilities and also Microsoft.CodeAnalysis nuget and it downloaded hundred of dependent System.* packages. Before it was only Microsoft.CodeAnalysis.Common, CSharp, CSharp.Workspaces, etc. What is happening? Does it work in Visual Studio 2017 with .NET Framework 4.7? Or is it included in 4.7 (I didn't find it anywhere)? Should I wait?

Older versions of Microsoft.CodeAnalysis.Common only had a variant for .Net 4.5, with few dependencies.
But since version 1.3.0, the package also has a .Net Standard 1.3 variant, which has many dependencies, as is usual for .Net Standard 1.x packages. When installing such package into a project targeting .Net Framework 4.7, NuGet installs all the .Net Standard dependencies. As far as I know, there is nothing wrong with that and the project should work fine.