How can we add webparts outside lms space eg. location other than /sites/lms and put them on main site, with context of courses that the student is currently sign to?

You should be able to do that by:

• going to Site Actions->Site Settings->Site Features
• and activating SharePoint LMS Web Parts Feature.

You'll then be able to insert web parts to your main portal.
If the web part you're looking for is not there, e.g. Gradebook, you can use API to create web parts.

Be careful: if items in the API reference have description of "This class and its members are reserved for internal use and are not intended to be used in your code." it means that this method, attribute and so on can be changed later by us.