@ex-pubserv
The answer is yes to the additional income. There are caveats - how much - how frequent - sole trader - VAT - company.
A company in the UK will be likely to be VAT registered. You need to tell us whether your income will exceed the VAT thresholds - if not no VAT, if yes then you will have to regsiter as a sole trader or a company. (Few issues to consider.)
As regards the EU funded public bodies that might be two distinct matters. 1 you got a gold plated deal and your departure package restricts you. 2. EU regulation - you could ask the Commission directly. These guys are far better at answering directly than their irish equivalents.
Actually there is a good bit in your query that raises more matters. It may be that you would benefit from professional advice.