Indirect political power? Well, all of the monarchies are rich as shit so…
Their ceremonial roles also tend to give them a lot of access to powerful people, and a lot of prestige.
Also the UK monarch literally removed an Australian Prime Minister (and dissolved the Aus Parliament) in 1975, and still has the power to do it again.
That was the governor-general, who afaik is only nominally under the control of the monarch. To all intents and purposes they are appointed by the prime minister and use their powers independently. I think that's how it works in all of the non-UK countries that have the British monarch as their head of state.
Anyway the current British royals only seem to be interested in using their influence for stupid shit like promoting homeopathy and interfering with random building projects (and protecting their own status and wealth of course).
No it doesn't.
They formed a government in January. The reason they didn't have one prior to that is because Sinn Féin pulled out of the power-sharing agreement with the DUP, mostly over the Cash for Ash scandal (which was as stupid as it sounds), and they failed to reach an agreement to restore it for a long time. Their system of government requires at least the two largest parties to form a coalition government - other parties can join it if they want to - and that has consistently been the DUP and Sinn Féin in recent years.
Also when the assembly and executive are suspended they're just ruled by the UK government in basically the same way that they rule over England. They aren't directly ruled by the monarch.