So far as I know, the incumbent Taoiseach has to go to the Áras to seek a dissolution of the Dáil. At least that’s what happened when the FG/Lab coalition fell over John Bruton’s budget in '82.
As for the Spanish situation, I’ve mixed feelings. If I had a vote there (and I’m entitled to one, just can’t be arsed making the effort to actually get one), despite not being what they claim to be, the PSOE would probably get my vote - PP definitely never would - so in that sense, I’m glad to see the PP out on their ear by any means. That said, both the Spanish electorate and PSOE Executive have failed to give Sánchez their support when it was sought, so the claim that he’s ruling without popular support or a democratic mandate isn’t without foundation, especially as he’s not even a member of Parliament.
What impact this has on the Catalunya situation will be interesting. The PSOE supported the PP’s stance on Article 155, so can’t credibly do a u-turn on that now. Against that, as they don’t have the numbers in the Cortes, they may need to strike a deal with Catalan Nationalists for their support. I just hope for the sake of everyone there this doesn’t result in the kind of stalemate that happened after the last two elections when the country effectively had no government for the guts of a year.