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On the original series of 3750s, there is not a special IOS (feature set) to support PoE nor is there an IOS per port count on the 3750 model.
On the original series of 3750s, there's an IP Base feature set (which comes with the -S suffixed models) and an IP Services feature set (which comes with the -E suffixed models). For a while there was also an IP Advanced Service feature set, which I recall needed another paper license to use, but Cisco in the latter IOS versions combined the features of both Services images into just one feature set with only one license requirement.
Also on the original 3750 series, you can technically run any feature set image on any 3750 model. The Service feature set, though, required a paper license. From the factory, for the -E suffixed models, you've purchased a Services license. For -S suffixed models, from the factory, legally you need to purchase a license to use a Services image.
BTW with the later -E/-X series of 3750s (don't confuse the model 3750-E with the license -E suffix), support feature licensing on the hardware. Again, the original 3750 series, including the G variants, are "paper" licenses. (Also from the factory, an original 3750 series would have an IOS image installed that matched its license.)
Is an IOS from a 3750-24TS-E compatible with either a 3750-48TS-S or 3750-48PS-S?
Technically, yes. Legally, no, unless you've purchased a paper license for the -S model.
Also do the switches use the same IOS regardless of if it is a 24 or 48 port switch with or without PoE?
Yes, IOS doesn't care about port count or PoE capability. (Note: for PoE, it's possible some very early IOS versions didn't support PoE [?] if original 3750s had no PoE models. However, if true, whatever IOS version that supported PoE would work with non-PoE models too.)
If the above seems contrary to what Leo posted, I suspect he also has 3750 stacking in mind. When you stack 3750s, additional rules apply to IOS actually installed on the devices. In general, same exact image on all devices is the ideal. Next best, is all stack units have the same IOS version, but with different feature sets. (In these cases, the stack master normally runs the "best" feature set, and its features becomes the stack's features.) Lastly, it's also possible to have stack members with different IOS versions, but not all different IOS versions are compatible. The latter is something to be really, really avoided.
The reason I mention the last, although legally you cannot install the Services images on either of your two -S models (unless you have separate paper licenses), you can stack them together and legally use Service image features as long as the -E model is the active stack master.
One last item, the original 3750 series allowed for lifetime IOS version upgrades. I believe Cisco changed that policy, though, later requiring a maintenance contract to upgrade the IOS (as is the situation for almost all their other hardware). As original 3750s are EoL, legally there may not be a way to upgrade either the IOS version or the IOS feature set. (What Cisco might offer is a trade-in allowance toward newer equipment.)
