I'm booking a flight one-way from Ashgabat to Dublin via Istanbul. If I book the two legs separately, both with Turkish Airlines, it'a about €120 cheaper than booking it as a single (again with Turkish Airlines, no idea why). Does anyone have an idea if they will still check my baggage through to Dublin, or will I have to collect and re-check in (I have time, but don't want the hassle, as well as the fact I'll probably have to get a visa stamp "on arrival").
Turkish Airlines has a web page detailing their transfer procedures. In your case:
International transfer passengers flight routes issued on single ticket continuing to a foreign destination with Turkish Airlines
Passengers disembark with only their cabin baggage and proceed directly to the boarding gates without any flight or passport procedures.
Passengers who have not obtained their boarding pass for the connecting flight at the initial point of departure must apply to the Turkish Airlines Transit Check-in desk.
International transfer passengers (flight routes issued on separate ticket) continuing to a foreign destination with Turkish Airlines*
Passengers reclaim their baggage from the international arrivals hall and proceed to the international terminal to apply to the relevant airline counter for check-in procedures.
So if you book separate tickets, you will need to collect your luggage and check it in again. (Seems like an acceptable hassle to me for €120.)
Note that Turkey does not do any passport checks for passengers in transit (staying in the international transit area of the airport), whether they collect their checked luggage or not. You only need a visa if you want to get out of the airport.
In general, booking separate tickets has the downside that if your first flight is delayed and you miss your second flight, the airline disclaims responsibility. With a single ticket, it's always the airline's responsibility to get you to your final destination. In the case of Turkish Airlines, their rules state:
If due to circumstances beyond its control Carrier cancels or delays a flight, is unable to provide previously confirmed space, fails to stop at a passenger's stopover or destination point, or causes the passenger to miss a connecting flight on which the passenger holds a reservation, Carrier shall (…)
I believe that a separate plane ticket (even with the same company) is not considered a “connecting flight” here, so with separate tickets you would be on your own. That may be worth paying €120 more.
You have two separate journeys, even if they are on Turkish Airlines. The second flight will not have your name as a passenger and thus your luggage will not be transferred.
In fact, your luggage will be tagged only till Istanbul when you check-in. As far as the airline in concerned, you only have a ticket to Turkey, so you need to make sure you are able to enter Turkey. Either you are exempt, or have applied in advance for a transit visa. This is because you will not be allowed to board unless you are able to terminate your journey in Turkey.
Once you land in Istanbul, you will follow the normal passenger that are going to Istanbul - you will follow the arrivals queue rather than the transit queue. You will most likely have to go through immigration, collect your luggage, go through customs and then enter Istanbul.
Then, simply follow the signs for international departure and check in again at the Turkish airline counter for your next flight. Hand them your luggage, this time they will tag it to Lisbon. Present your passport and flight reservation, get your boarding pass, go through security, immigration and then board your next flight.
Make sure you have sufficient time between flights to complete all the formalities and account for any delays.