How would you address an invitation to the Irish ambassador?
A foreign Ambassador accredited to the Court of St James's is accorded the style of 'His/Her Excellency' within the United Kingdom and Colonies.
A social invitation to the Irish Ambassador would simply be inscribed 'The Irish Ambassador'. If you are also asking his wife this would read 'The Irish Ambassador and Mrs XYZ'.
The envelope, however, would be addressed to 'His Excellency Mr John XYZ', followed on the next line by 'The Irish Ambassador'.
What is the correct wording in full when giving the Loyal Toast?
The Loyal Toast, as approved by The Queen, is 'The Queen'. It is incorrect to use such forms as: 'I give you the Loyal Toast of Her Majesty The Queen'. To obtain the necessary silence, the Toastmaster (or another) may say, without preamble, 'Pray silence for your President', who will then propose the Loyal Toast.