Charlie Brumbaugh gave you excellent information about land turtles. It may be exactly the same for aquatic species, but I'd really like to provide you with information about those. Unfortunately, it's turning out to be more complicated than I thought. If you don't mind, I wanted to at least post something, even though it's not exactly what you need yet. I'll probably have to make some phone calls to experts in order to get fully accurate information. I promise to come back with reputable sources, or delete it if it's just taking up a lot of space without really being helpful.
If some kind soul comes along with the right answer, that would be great!
The closest thing I've found so far pertains to the Western Painted Turtle, (Chrysemys picta belli). According to this fact sheet, an early thaw can be dangerous indeed.
Western painted turtles survive the winter by hibernating. In the fall they put on extra fat and, as the temperature drops, they gradually become less active. Finally they burrow deep into the mud at the bottom of ponds and go into hibernation. Scientists have found that the blood of hibernating turtles actually changes. Like the antifreeze used in the winter to keep the water in car radiators from freezing, turtle blood changes so that it can withstand cold temperatures. As a result, turtle body temperatures can drop to only a few degrees above freezing -- much lower than that of most animals that hibernate. As they warm up, turtles wake up and slowly become active.
Early warm spells can be dangerous because if they wake up too soon, a sudden return to cold weather may catch them unprepared and they may freeze. In fact, winter weather can be the biggest danger a turtle faces in its adult life. Painted turtles are hardy though, and it is not unusual to see them swimming under ice during the winter.
That being said, you don't actually have the Western Painted Turtle in Ohio. What you have is the Midland Painted Turtle, Chrysemys picta marginata. According to Ohio DNR Wildlife:
Midland painted turtles are among the most abundant and most conspicuous turtles in Ohio. They are particularly fond of basking and can be seen by the dozens on logs and along the banks of most bodies of water through the summer.
There are several subspecies of painted turtles in the United States, but only the midland painted turtle occurs in Ohio.
To confirm, here's a list of Ohio's 12 species of turtles. In the description of the midland turtle, it says
For more information, check out the Wikipedia page on Painted Turtles. Although this Wikipedia article discusses several varieties of painted turtle, only the Midland Painted Turtle lives in Ohio.
From that page:
While hibernating, the body temperature of the painted turtle averages 6 °C (43 °F). Periods of warm weather bring the turtle out of hibernation, and even in the north, individuals have been seen basking in February.
Unfortunately, it doesn't go on to say what happens if there's another freeze after that time.
You do have a Red-eared slider (Trachemys scripta elegans), and two varieties of map turtles, the Common Map Turtle (Graptemys geographica), and the Ouachita Map Turtle (Graptemys ouachitensis). So far I'm finding different requirements for the Maps, which may indicate a better chance of success. I need to do some more research, though.
EDIT: Today, February 27, I spoke with a very kind woman at the Ohio Division of Wildlife. I told her where you live and which turtles you've seen. She said all your turtles have adapted to these changes in weather, and can go back into whatever state of hibernation is necessary to protect themselves. A full hibernation isn't always necessary, especially if it's just for a few weeks or so. Some Map turtles tend to end their hibernation earlier than the other breeds anyway. Unfortunately, if the turtles haven't sensed the weather getting colder, and are spending a lot of time out in the sun, a surprise freeze will cause them to pass away before they can get back under water. She said it's very rare, and happens quickly, so hopefully they don't suffer.
Those who are deceased can be seen at the water's edge, so it's up to you if you want to look around the pond area. I can't stand to see deceased creatures, so I'd probably stay away, but I won't even step on ants, and have been called over-sensitive many times!
I'd appreciate if you keep us posted, if you want to, and give us an update. I sincerely hope Mappy and his friends will be fine!