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Lines and polygon dimensions are never exact and subject to the limitations of the feature class resolution and tolerance. Infinite precision is not possible and as a result these settings tell GIS to consider measures below those levels as "close enough" to exact measurements. There is no setting for absolutely no tolerance and infinite resolution for ...
As commented by @KHibma: Unless you've created a package to reference the data, it'll make a copy of that data for each package you made. Thus one edit/change won't impact another package. When you open the package, check the source of that data. A key thing to understand with map packaging is that the map package contains a copy of the source ...
First, use graphics from the Draw toolbar and use the polygon to draw what you want. Then right click on the Layers on the TOC. Scroll down to Convert Graphics to Features... Save your new polygon as a Area of Interest or whatever you like to name it. Use Extract by Mask under Spatial Analyst Tools --> Extraction.
Esri now has a tool for this in 10.4 (hooray): Recalculate Feature Class Extent. I was running into this issue when I was creating a feature class and writing geometry into it with arcpy. Apparently those tools do not update the extent (probably a good idea for performance reasons). I have been successful resetting the extent in 10.2.2 with @Lou 's ...
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