How do I export data of the specific question sets used for 1-to-1’s (including the individual responses) ?

This can be exported via the ‘Question sets’ tab:

With 1-to-1's:

In the top navigation bar, click on Assess
Select 1-to-1
Click on the Questions tab (far right)
Find the specific 1-to-1 Question set
Click on the three pips
Select View report
Click on Download individual responses

With Objectives:

In the top navigation bar, click on Assess
Select Objectives
Click on the All tab (far right)
Select whether you want to see Past, Current or Future Objectives
Add the desired filtering
Select Actions
Click on Export CSV

Is it possible to export just the 'Reviewer' and 'Reviewee' feedback for a specific 1-to-1 and NOT export all individual responses?

Not at the moment - Although, you can follow the steps above as this information will be included.
Then you can separate the data in the CSV directly.