I used the trains from Dublin to Ennis and back in July and they were just fine. The main line train was pretty slick and the Limerick Junction to Ennis branch line was fine. The WiFi on both worked amazingly well, for what that is worth! You can buy tickets online in advance quite easily as well.

I have driven in Ireland and it can be quite exciting. Be careful about the insurance though. This is all well covered in the other threads.

There are no LCC flights from LHR to Dublin (or anywhere else, really).
Aer Lingus fly from LHR Terminal 2, British Airways from LHR Terminal 5. As both airlines belong to the same parent company there's a lot of codesharing.
LCC flights from London to Dublin are all by Ryanair and leave from London Gatwick (LGW), London Stansted (STN) and London Luton (LTN).
Aer Lingus also fly from LGW and from London Southend (SEN).
British Airways also fly from London City (LCY) to Dublin, as do Cityjet.
So lots of flights and if you're stopping off for a day or so in London, lots of choice. You would need to figure in the cost of getting to the airport both in money and time.

Perfectly doable without hiring a car,especially if you like to booze.Particularly if you like to booze the Irish way.
Dublin is fine for a night or two but no more.
I'd recommend getting out into the country,maybe with a train to Cork or somewhere like that.

www.irishrail.ie
www.buseireann.ie
www.paddywagontours.com

How about starting on the west coast?
 

If you can withstand a 757 for 6.25 hours, fly direct from Newark to Shannon on the west coast. Drive to the Cliffs of Moher (wear a warm windbreaker and scarf!) through the Burren and then up to Galway (plenty of pubs and music). Next day drive through Connemara to Roundstone (although probably really quiet in January) and to Clifton and maybe Kylemore Abbey if that interests you. Return the car in Galway and take the 2.5 hr train to Dublin the next day. The highway to Dublin is fine, but driving into Dublin city is stressful.

Or drive from Galway to see Newgrange and then to Malahide for the night. Or to Howth for dinner at the Abbey Tavern and stay at King Sitric. Easy drive to DUB for a flight to London.

If you don't want to drive, contact www.silverbus.ie and they can customize a transfer to Galway via the Cliffs of Moher. You can take a Connemara day trip from Galway the next day. Check viator.com for tours. Then train to Dublin the following day.

The Cliffs of Moher and the Burren are certainly worth a visit, though they may be a bit bleak in winter. There is a regular bus to Galway via the Cliffs but there would be the problem of what to do with your luggage if you got off unless you booked an overnight stay. I image the boat trip, which was quite exciting in July, does not run in January!

The seaplane museum in Shannon has been well reviewed on FT if you are into that sort of thing.

You should be able to connect with a rental car from a train station if you want to start in Dublin but not drive. There is an online trip planner for public transport: http://www.transportforireland.ie/