In Denmark, the answer would be no for all those countries.
However, when submitting the quarterly/semi-annual VAT to the authorities, one must also register the amount of non-VAT sales (EU and non-EU).
Yes, the Freelancer can buy just the same as a private person, and later decide this is a business-expense. Recipient must be the Freelancer himself though, not a friend etc.
The invoice has to fulfill all requirements of a regular invoice under German law, see here (German)
The usual cash register-printout (Kassenzettel) does not suffice, at least not ...
There are a number of things an invoice must contain to be valid.
Your government has some info on this.
Your invoice must include:
a unique identification number
your company name, address and contact information
the company name and address of the customer you’re invoicing
a clear description of what you’re charging for
the date of the goods or ...
The way I understand it VAT number in EU serves as EU-wide tax id. There is a VAT number database where you can check the validity of counter party VAT number. In Czechia you are required by law to acquire VAT number no longer than a month after you issue your first invoice to an EU counter party. I suggest you check your regulations as you might be required ...
As you are supplying services from UK to an EU country you may need to check with that country and register there for VAT even if it is for a pittance . In this case UK threshold should not be your yardstick (except when the payments impact your turnover)
You need to ensure that this IS a one off cross border service (no supply by you of physical materials ...
The gov.uk website has some useful information on how to invoice customers and what information needs to be included on an invoice. See https://www.gov.uk/invoicing-and-taking-payment-from-customers.
Invoice numbers need to be unique across the business, not per customer:
You must include ... a unique identification number
There are some additional ...
I have no knowledge of VAT (because I'm US-based and don't need it).
However, customarily every invoice has a unique number - regardless of client/customer. You should never have more than 1 invoice with the same number, ever. This allows you to find the exact transaction at a later date. Having multiple invoices with identical numbers will inevitably ...