Irrespective of the platform, you just need to consider two important things while importing data.
You have the required fields available. E.g., if you have validation rules on a field or that a field is marked as required at API level, make sure you have those in the load file
And that you have the "correct" data populated in there. E.g., if you are updating a data in a numeric field and that you are sending say string, it will fail
As long as you have these two things taken care of, your loads will work fine.
As for your questions:
I need to know if my csv file can only contain the data that needs to be mapped and the record ID. (i.e., If I exported extra data that doesn't need to be replaced/mapped, can I delete it from the csv file?)
Yes. As long as it isn't really is required, you can always remove those.
Second question is related to the record id -which for me is the opportunity ID. I'm a little confused on which field to map this to. Would I map it to the opportunity name? (or will that overwrite the name and put the ID in it's place?)
No, it does not map to the Name field. Its the "Record Id", the standard Salesforce Id of the record. So if you are updating records, the Opportunity Id is the value that is used to find the respective record and make necessary updates on all other fields. Remember, the Id fields are read only, i.e., you cannot update it. And that you only require it in update scenarios as the Ids are created net new whenever you insert records.
I need to move data from an existing text field in opportunity to a newly created picklist in opportunity.
In your case, you just need to make sure you map the old field values to the newly created field, use the Opportunity Id and perform an update and that should work.
Here are few resources where you can get more details around this topic: