"...We are also told that they sang rebel songs as they sailed. Did they?..."
Doubtful
1) They were all volunteers (IIRS They were the 10th Div which was all Regulars, so they were LONG TERM Regulars and not "Hostilities Only" soldiers like, say the 29th Div).
2) Such action would have been classified as "Conduct prejudicial to Good Order and Discipline" and therefore a military "crime"[1]. Had they been 16th Division, there is a possibility, as these were formed from the "National Volunteers" (in the same way that the 36th Div was formed from the UVF), but I think both of these Divisions were still in training at this time.

"....fighting for ''the wrong country'' and ''the wrong cause'' ..."
I think this implys that they should have been fighting with the "Irish Republican Brotherhood" (forerunners of the IRA) instead of following John Redmonds call to enlist and get the war over (the arrangement was that Home Rule would be initiated after the war was over[2])

"...My guess is that those involved ... would have had a fairly low opinion of any ...who took up arms against the very army in which they were fighting ..."
I seem to recall reading that the "Easter Rising" was not as popular at the time as legend would have us believe, coming at a time when many of the families of the dead of the Royal Dublin Fusiliers and the rest of the Division were still in mourning from Gallipoli and other Irish men were on the Western Front with a major offensive in the offing.

"How sad that Dublin for so long refused to recognise their service and sacrifice"

Indeed

10,16,36 Irishmen all.

Walrus

[1] Not a criminal act, but an offence entered on the soldier's record, thus affecting possible promotion, good conduct pay etc. [2] Remember this war was going to be "Over by Christmas".