Should I bring my iPhone5? Sprint says they can unlock it for me...

Some iphone 5s are Gsm,some aren't, I guess they ran the. multidigit model number thru Apples checker.
Double check the frequency bands it has, you want 900/1800mhz for europe.
Merely unlocking without the above frequencies is of no benefit.
Its so confusing, because some iphones are the US only system some have GSM but only US. frequencies, and some the worldband ones.

Are they charging for making it unlocked?
Turning off the data connection does not effect TXT.

If you can get it unlocked, do so. It will increase the value of the phone should you ever decide to sell it. You can check eBay to see the difference in prices for unlocked vs. Sprint phones. After it's unlocked, put a Three sim in it in Ireland. Plenty of data, calls and texts for £20 good for 30 days. Then use it as normal.

"I thought my phone was not GSM and would not work internationally but Sprint says they have a group that can unlock it for me so it can use a SIM card."

Whether a Sprint phone will work in Europe depends on the model. The Sprint iPhone 5 is a "world phone" and has both CDMA for North America and GSM (with a SIM) for other places. So yes, you can use it with either the Sprint SIM (at Sprint's international roaming rates) or a local SIM (at local rates - much cheaper).

"If I do this, I think I would still want to disable data services, is that right? Because those are the ones that add up so fast?"

Yes. However, you can usually buy an international plan that will reduce the cost. This can be easier than buying a SIM locally, and if you're not using it that much (say, just for emergencies), can work well.

"And if I disable data services, would that also prevent me from sending texts? I think so... I do want to be able to send texts."

No, regular texts go over the voice network, not the data network. Disabling data still allows texts.

"I was originally planning to buy a cheap phone and sim card over there, but am now not sure it's necessary. Would welcome advice please."

If you can make sure your phone is unlocked, there's no need to buy a new phone in Ireland. Just bring your current one. You should investigate the rates for Sprint's international plans, so you can decide if you want to use that, or buy an Irish SIM.