This is not a rail-to-rail opamp. Hence, if you feed it +/-5V, its output can never swing to +5V; it'll go to about 3.5V max according to its datasheet. So your observation of 3.7V out is about right. I don't know why your first stage manages to swing all the way down to -4.3V; I wouldn't have expected it to go below something like -4V.
No. It's not rail to rail either. Furthermore, it has a fairly high input offset current which is far from ideal in your use case.
Try something like AD8605/8606.