This UK govt page lists the requirements to apply for a new National Insurance number. The first item is this:
You can apply for a National Insurance number if you live in the UK and have the right to work in the UK.
Thus, you'll have to wait to apply until after you've actually moved to the UK.
Summary (Jan 2022): Provided you meet all other conditions imposed by your GAE visa, you will be able to leave and re-enter the UK as many times as you wish during the visa validity.
As of the time of the answer, neither the Government Authorised Exchange (GAE) visa information page nor the associated caseworker guidance has explicitly mentioned that one is ...
Based on the government's website the safest thing is to go through the red route, as you'll be travelling with belongings worth more than £390. If you already filled in your Tor1, then make sure you show it if asked.
How to go through customs
What to do when you arrive at customs
In most cases, you’ll be able to go through the green ‘nothing to