This is actually simple in its complexity and in part has to do with a concept called "Irish republican legitimatism"

In 1916 a small coalition of Irish nationalist insurgents seized control of buildings in Dublin and proclaimed an Irish republic, this was done by aptly reading out a proclamation on the steps of the Dublin general post office. The proclamation's most important line for here was,

"Until our arms have brought the opportune moment for the establishment of a permanent National Government, representative of the whole people of Ireland and elected by the suffrages of all her men and women, the Provisional Government, hereby constituted, will administer the civil and military affairs of the Republic in trust for the people."

This in general meant that Ireland would controlled by a non-popularly body until such a time that it could elect a proper government.

Later in the 1918 general election Sinn Féin which had become the party of the Republican movement swept the polls and took 73 of Irelands seats in the UK house of commons, it then took this opportunity to declare itself an independent government representative of the popular will of the majority of Ireland. This 'First' Dail (Dail meaning assembly) is part of Ireland governmental continuity even until today despite not being elected as members of the Dail or even to an independent or semi-independent Ireland. The First Dail however was fully aware that the UK would seek to sweep it aside after it had refused to acknowledge it and so the Dail passed legislation that in the event that the Dail was unable to function then the military would be legally obliged to take control until the Dail could be reformed.

The army of the Irish Republic was at this point the Irish Republican Army, or in Irish the Óglaigh na hÉireann. This is a tad confusing as Óglaigh na hÉireann does not mean Irish Republican Army(it should be something like Arm Poblachtach na hÉireann if my Irish isn't too rusty), it more loosely translates to Volunteers of Ireland and was the Irish translation used originally for the Irish Volunteers, the group that launched the 1916 rising. Later when the Irish Republic would establish its defence forces, they too would be called the Óglaigh na hÉireann. The IRA, including the PIRA would still use the Óglaigh na hÉireann title and refer to themselves as such as well.

As the Irish war of independence slowed down a new election in 1921 was held and Sinn Féin won again, interestingly Sinn Féin actually lost massively in the popular vote as 124/130 of its seats were taken unopposed. This 'Second' Dail then passed the Anglo-Irish treaty 64 to 57.

Another election was called in 1922 and was primarily fought over the issue of the Anglo-Irish treaty, the pro-treaty section of Sinn Féin won narrowly and the anti-treaty section launched the civil war, the pro-treaty IRA became the National Army then the Defence Forces while the anti-treaty IRA retained the IRA name.

The anti-treaty TDs of the second Dail formed the Republican government, essentially an opposing government to that of what would become the Free State of Ireland. It was weak and could not exercise much authority as while it took control over the bulk of the IRA the vast majority of the civil authority sided with the free state.

The civil war was a short but bloody affair with the free state winning firmly albeit with the lose of key figures like Michael Collins. After the war Éamon De Valera, the president of the Republican government who had been elected by the army executive of the IRA led anti-treaty republicans to taking power electorally. Most would think this would be the end of the issue however for De Valera to go into government he had to do so in a new party as he could not convince the militarists to support the electoral endeavour, regardless relations between Fianna Fáil (De Valeras new party) and the IRA remained cordial for the time especially as the IRA helped counter Irelands fledgling fascist movement in street brawls.

The second Dail also had a bit of an oddity, it wasn't actually dismissed properly. Due to the treaty setting up a new government it was decided that the second Dail would hand over power ceremonially to the third to keep continuity, the meeting however never happened as the civil war broke out. The free state claimed continuity regardless while anti-treaty forces denied it.

This means that in the eyes of the anti-treaty side the second Dail was still in session and the legitimate government not the third Dail and its successors. However, the people that the anti-treaty side acknowledged as valid members gradually reduced over time. The 64 that voted for the treaty were struck off immediately for violating their oath to protect the Republic(a requirement to take their seats), others were struck off for taking seats in successive elections, especially as to take your seat in the free state you had to swear loyalty to British Crown. By 1938 only 7 members remained, ironically they would have been struck off themselves as they had ran in past elections but were unsuccessful, these 7 in turn invoked the first Dail's legislation in regard to the Dail being unable to operate to pass their authority over to the Army Council of the IRA.

To take stock before the 1969 split in the IRA, depending on your interpretation of the law and oaths, the chief of staff of the IRA Army Council was not just the head of the military but also the head of the Irish state. The arguments in favour of this position are that, the first Dail allowed for the army to take control, the army of the Republic was the IRA as set out by the first Dail, the IRA had sided majority with the anti-treaty side, the ratification of the Anglo-Irish treaty was illegitimate as the supporters violated their oath of office, the majority of "valid" TDs of the second Dail which had not been properly dismissed transfered authority to the army.

The PIRA would slowly drop its position of being an alternative government of Ireland, fully dropping it by 1986 but in 1977 Seamus Twomey was a claimant to the position of head of state.

Fianna Fáil always feared a resurgent Sinn Féin, its why its subtitle is "the republican party", the IRA wasnt just fighting for the unity of Ireland it was also an existencial threat to Ireland's institutions and it was genuinely feared that the an IRA victory in Northern Ireland would see them turn their guns southward to settle old scores.

Whilst we wait for someone to address your specific questions I thought I would link some previous answers that touch on the role of the Irish government during the Troubles.

In the first part of u/thefeckamIdoing 's answer they talk about the complicated relationship between the Irish government and the original IRA during the first years of the Irish Republic/Free State. It's important to note that by Seamus Twomey's time the organisation that had fought in the War of Independence 1919-21 had splintered several times, with Twomey belonging to the Provisional IRA.

This thread is about collusion between the Irish government/security services/public officials and the Provisional IRA during the Troubles. The answer by u/charliesfrown is most relevant to you. They say that Reunification was not a high priority for the Irish government and although there was some supporters of the IRA in the Republic who were willing to help them, overall public support was low.