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Another example of what i'm trying to do is say I have a page in sharepoint with a calendar part and I want to show that same calendar in my external web application. 
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Showing Same calendar web part is not possible in a non-share point site . But you can try to &nbsp;build a calendar in asp.net platform and sync it with your SharePoint calendar list &nbsp;I mean get the information from calendar list and it require lots of work. If you are a developer you can try this 
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Is it possible to grab a webpart hosted in sharepoint and reuse it in an external web application? &nbsp; (e.g. If I have a KPI webpart on my sharepoint page, but also want to include that same result on a web page in my asp.net web application?
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Is it possible to grab a webpart hosted in sharepoint and reuse it in an external web application?   (e.g. If I have a KPI webpart on my sharepoint page, but also want to include that same result on a web page in my asp.net web application?

Another example of what i&#39;m trying to do is say I have a page in sharepoint with a calendar part and I want to show that same calendar in my external web application. 

Showing Same calendar web part is not possible in a non-share point site . But you can try to  build a calendar in asp.net platform and sync it with your SharePoint calendar list  I mean get the information from calendar list and it require lots of work. If you are a developer you can try this 

Use SharePoint WebPart in external web application

Microsoft Sharepoint is a software platform and family of software products used for collaboration and web publishing combined. These capabilities include developing web sites, portals, intranets, content management systems, search engines, wikis, blogs, and other tools for business intelligence and collaboration. SharePoint has a Microsoft Office-like interface, and it is closely integrated with the Office suite.

Microsoft SharePoint

The .NET Framework is not specific to any one programming language; rather, it includes a library of functions that allows developers to rapidly build applications. Several supported languages include C#, VB.NET, C++ or ASP.NET.

.NET Programming

The successor to Active Server Pages, ASP.NET websites utilize the .NET framework to produce dynamic, data and content-driven web applications and services. ASP.NET code can be written using any .NET supported language. As of 2009, ASP.NET can also apply the Model-View-Controller (MVC) pattern to web applications