Did they trade?
Did Rome try to convert any Irish to Roman religion?
Why wasn't Ireland seen as important while Britain was?
Tacitus writes that at one point the general Agricola states confidently that a single legion could probably subdue the entire island. He mentions that it would be a good move for breaking local morale, and might possibly be worth subduing as it served as a potential launching point for native attacks across the sea. Given this concern, it stands to reason that various tribes would have been communicating with each other.
Barry Raftery argued in the 90's that the town of Drumanagh served as a trading ground between Ireland and Roman-occupied Britain. Citing archaeological evidence, he asserted that the island was inhabited by Irish, British, and Roman persons. With trade comes talk, and so there would have certainly been communication.
Ptolemy's Geography had a pretty solid map of Irelands coasts and ports - information he claimed to have gotten from local merchants who traded with the island frequently for various good. Tacitus also notes Irish wolfhounds being brought to Britain. There seems to be evidence for pretty ubiquitous trade between the islands.
I can address your second question I believe.
For most of the time the Romans were in Britain, they were not Christian. Christianity as a state religion didn't occur until well into the fourth century. The Empire prior to sanctioning Christianity was not terribly interested in converting the peoples they conquered. As polytheists, as long as the power structure was given what they perceived to be its due, they were generally content to live and let live with respect to indigenous deities, rituals, and festivals.
Polytheists can be quite adept syncretists and have little trouble reconciling and fusing various belief systems. These Romans wouldn't have seen any reason to attempt conversions.
That changes in the fourth and fifth centuries. The Edict of Milan in 313 stated, "that it was proper that the Christians and all others should have liberty to follow that mode of religion which to each of them appeared best." That's just a legalization of the religion. It didn't become the official religion of the empire until Emperor Theodosius in 380.
As the Empire is collapsing in the fifth century, St. Patrick (and others) enter the scene and we begin to see significant effort at converting the Irish to Christianity...and that leads to an entirely fascinating episode of history where the Irish and Roman churches increasingly begin to clash leading to the Synod of Whitby in 663 ... blimey, I could go on. :-)
I am fascinated by your third question as well. I hope to see more answers.
The literary evidence mentioned here is somewhat controversial. It is true that Roman material does show up in Ireland, but the amount there is minuscule, an it is widely debated and it is widely debated how to interpret its transfer. Did it arrive by trade routes, carried by merchants seeking profits? Did it arrive by a continuation of pre-Roman gift exchange between Irish and (now Romanized) British elite? Were they diplomatic tools by the Roman state? Were they the booty of plunder raids?
Every one of these possibilities have ample comparative examples in other areas in contact with the Roman world, and so the point is rather moot. There are two real problems here: one is that in general this period of Ireland is extremely poorly known compared to preceding and succeeding periods, to the point where in the early nineties a well respected archaeologist could say that we knew there were people in Ireland then, we just don't know where. The situation is better now, but there are still very fundamental gaps in the archaeological record. The second problem is somewhat more basic and related to my particular perspective: fired clay amphora, which are the great markers of Roman trade routes, were largely not used in Britain, rather wooden barrels were. Amphora fragments survive, wood doesn't, and this means that the fundamental categories of traceable commodity exchange--wine, oil and garum--are simply much less visible.
Archaeology in Ireland is rapidly expanding, so things are picking up and I have great hopes for what may be revealed. But for now it is all very murky. My main source is John Soderburg's "Between Britain and Spain" in Rome Beyond it's Frontiers.
Well, we know the Irish were raiding Briton and taking slaves during the 4th and 5th Centuries.
This was how St. Patrick was captured. He was enslaved by Irish raiders at the age of 14, and sent to tend sheep out by himself in the field, working for a Druidical high priest. After six years a slave, in 407, he heard a voice telling him to go home. He walked to the coast, and found a merchant ship willing to take him back home to Britain, thus indicating that such trips were made.
Along the way home they were blown off course and landed somewhere in northern France, right after the barbarians had blown through, and they found the place a "wasteland", where they almost starved to death from lack of food. (As they were expecting to find civilization there, which had been obliterated.)
Patrick underwent clerical training and returned in 433, having followed several other Catholics to the island.