"Was the Irish Free State founded democratically?
The Irish State was formed though a war of independence from colonialism which lasted centuries; as were many countries; do you have problems with this or do you think they should all have stayed in the Empire?

"….was established by agreement of all three Governments involved."
It was established reluctantly, after centuries of unrest. The negotiations for a treaty followed a period of state terrorism by British troops specifically put there to create a situation where the best results could be obtained(for a British Government and an extremely belligerent Protestant minority).
The irony was that the treaty eventually signed was opposed but all but a small minority of the population of the six counties. The Catholic minority were in favour of a totally united Ireland (either immediately or eventually); around half the Protestants were totally against any part of Ireland being under British rule; the minority remaining were prepared to go along with the partitioning. As I said, the treaty was forced through and maintained by violence add threats of invasion.

"What threat of invasion?". "If you don't sign this treaty granting you independence we will invade you!! "

Exaxctly;
"The Prime Minister (Lloyd George), of course, needed more than this: all must sign; if they did not, he solemnly promised that he would not even give them time to lay the matter before the Dail: it would be "war within three days," and war more terrible by far than any they had yet experienced. At 7:45p.m. The meeting broke up. Griffith had agreed to sign; Collins appeared to hesitate only over the Oath; Barton, who came to this meeting as a sort of host curiae, had not committed himself at all."
From - 'The Damnable Question – 126 of Anglo-Irish conflict'. George Dangerfield, 1976.

" Catholic population of Northern Ireland has grown and has grown steadily since May 1921, whereas, the Protestant population in the Irish Free State and then latterly the Irish Republic has plummeted."
There has been no evidence of persecution of non-Catholcs in the Republic, despite the Catholic Church having a strong influence (no longer the case as the church has lost its credibility following the clerical child abuse revelations).
On the other hand, there has been anti-Catholic repression and violence throughout the six counties. The stated aim of the original Home Rule Bill was to set up a Protestant State where the Catholic third of the population had no say in the governance of the state.
This has lasted throughout the state's existence and culminated in the peacefully held Civil Rights Marches in the late '60s which were responded to by baton charges by the RUC and the directing of the marchers through stone-throwing Unionist mobs – the next twenty-odd years speak for themselves.   

"Examples of the use of such force please."
You got 'em;
Members of my own family were burned out of their home in Derry by rioting Unionist mobs, forcing them to leave and flee to Dublin with their child in arms.

"Could one relevant factor possibly be because they did actually form the majority of the population perhaps?"
Even if the wishes of the majority had been met by the signing of the Treaty, which it wasn't, in a democracy it is the duty of any government to represent the interests of all sections of the population, minority or majority. This has never happened in the six counties and the question of 'power-sharing' has been brought to the conference table at gun-point and has yet to be resolved satisfactorily – not the basis for a democratic or stable state.
Anything else?
Jim Carroll