I have a Lookup Relationship field (Opportunity) in Cases to link the two together. There is lots of technical information that we need in order to troubleshoot the problem so I would like to display that technical info from the opportunity in the cases. Whats the best way to do this?

The solution to this requirement depends upon several factors:

1. How many fields from the related Opportunity do you want to be displayed on the Cases. If its just a couple of fields, you can use cross object formula fields on Cases to display the same fields inputs from the Opportunity associated to the Case. Also, users will not be able to edit those fields on the Case record.

2. You may want to explore using Service Cloud console (in case you have using Service Cloud as your environment). Using this, you can setup your Console views in such a way so that the Case record opens in one view and the related opportunity opens up in another view (you can also customize the Console layout of the Opportunity to show only those fields that are relevant to the support rep.

3. In case you do not have Service Cloud console and also if the number of opportunity fields is fairly high, you may want to explore using a VF page to be displayed as an inline component on the Case layout. This VF inline component can have all the fields that you need from the Opportunity related to the Case in a Read only mode.

Also question on Lookup Relationships, is it actually building a physical link with ID's from the one object to the other? Or is it just putting the opportunity name in there?

Yes, when you do a lookup, it is actually establishing a physical relationship between 2 objects (tables) and the underlying record id's. Using this relationship, you can have multiple actions like cross object workflow field updates, validations, triggers etc.

Hope this helps! Do click KUDOS in case the post helps you. Let me know how it goes