Not too demanding, it seems. GPS is supposedly a serial interface, too.
The only interface you won't find with all MCUs is ethernet, so this seems your main selection criteria.
A F4 device could suffice, there are several eval boards available to test.
I personally would start with the F4 discovery (+ BB extension), because it has an SPL-based demo webserver, not tainted by CubeMX.
Keep in mind that Ethernet is critical - the "feeble" peripherals in common MCUs cannot buffer any signifcant number of frames, and will easily be overwhelmed in an average office network.
If you want to dabble in ST's CubeMX is your decision. But read up about it before.
From my experience, the individual requirements are not too difficult to implement, several examples are available. But balancing interrupt priorities and DMA streams to keep everything going and balanced will cost you some time and effort.
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