Please advice .Net Run time required both ArcGIS Desktop & ArcGISEngine ?
No the ArcGIS Runtime is completely stand-alone and does not require Engine or Desktop. All you really need is the NuGet package from nuget.org and you're all set.
I installed ArcGIS Engine 10.6.1 but we previously using ArcGIS 10.0 and implemented the .net application .
If my application will work ArcGIS 10.6.1 version without issues?
Please do the needful.
Could it be that you're confusing the Engine SDK with the Runtime SDK? (these are two completely different products).
The runtime doesn't work with any version of ArcMap or ArcGIS Pro. It's its own thing and runs completely stand-alone.
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