I plan on travelling in Europe before/after my stay in Ireland – do I have to have return airline tickets directly from US to/from Ireland?

No, but you do have to show travel tickets that indicate when you will arrive Ireland (from the US or another country) and that you will be leaving Ireland within 12 months of arrival (to the US or another country).

If travelling to/from a country other than the US, you must make sure you meet the visa/entry requirements for that country.