Can you claim the costs from applying for a company number and vAT activation as expenses?

Yes, you can claim all business expenses. So all start-up costs, but also the costs of Dexxter, your social security contributions, your mobile phone subscription (partly), a desk... Take a look at the blog, where you'll find lots of inspiration for what you can deduct.

Normally, you should enter your costs under the correct quarter.

My start-up costs were invoiced and paid in quarter 3.
But in this specifically specified case, I started my company on 1 October (quarter 4).
Quarter 3 should therefore in principle already be closed and reported.

Do I still enter these start-up costs under quarter 3?
Or do I enter them under quarter 4 (with a special note/procedure/etc. if necessary)?

And since the vAT quarterly return from quarter 3 has already been closed... should I uncheck the box 'Include in vAT return'?
I will then no longer be able to recover the vAT, but I can still enter the professional expenses in the accounts for quarter 4.