Kevin,
I am unclear as to what records it is that you are seeking. Can you clarify?
If you want to find out where your family lived, I’d suggest you look at the marriage certificate I have found plus the 4 births. That will give you addresses for each of the events (plus information on John & Ann’s fathers, eg John’s father’s occupation which you said you were unsure about). Ann Shevlin’s address will be on her marriage certificate. With the addresses, I can try and find the various properties for you using on-line sources.
As far as Northern Ireland is concerned, most of the major sets of records are centralized and kept in PRONI (the public record office) in Belfast, but some of the others eg RC parish records, and the statutory births, deaths and marriages, are on-line. You already have the link for the births, deaths & marriages. Parish records are on:
http://registers.nli.ie/parishes
Each county has a library with a local studies section and that’s worth visiting for contextual material eg there might be information on trades such as carpentry, or a history of the town/townland where your family lived. But they generally don’t keep family material as such, save perhaps on some of the big landowners or industrialists.
Regarding the likely point of departure, there were sailings to America from Londonderry, Belfast, Queenstown (Cobh today) but by far the biggest port of departure was Liverpool. Far more migrants left Ireland via Liverpool than ever went directly. (Liverpool was a sort of European clearing house, and there were daily departures there as opposed to monthly from ports in Ireland). Someone leaving Tyrone in the 1890s would have jumped on a train to one of the Irish ports I mentioned. Then either sailed from there or caught an overnight ferry to Liverpool.