"I am assuming Nicea was an ecumenical council."
It was. Its where the Creed comes from, at least most of it.
"Who decided who should go and who couldn't go to Nicea?"
All the Christian bishops were invited because the main issue at hand was dealing with Arianism which was being preched by more than a few bishops.
"How was the council of Nicea paid for?"
I haven't a clue. Maybe at second collection at Sunday liturgies? A tax on goats?
"Who opened the council and presided over it?"
I believe the Emperor Constantine may have presided over the opening session. I know he called the Council. At least after the first session, it was presided over by +Hosius, the Bishop of Cordova and +Alexander, the Pat. of Alexandria. Interestingly, Constantine had very definite personal Arian leanings and sympathies (though he recognized the trouble Arius was causing in The Church) but the Council came down firmly against Arianism.
IIRC Constantine made a big speech at the opening but did not preside (there's 4 folks who may have presided). There's no record of Constantine presiding, and he wasn't even present after the first day.
I beleive Constantine offered to cover the travel expenses of all the invited bishops.