Assuming a Cisco 3825 router, if the Cisco SCCP phones on the network are using G.711 µ-law and are traversing a CUCME system connected to a SIP trunk that supports G.711 µ-law, is a PVDM2 DSP card required since the codecs are the same? Does having a Unity Express module installed on the system require a DSP card?
You should be good without DSPs. The major use of DSPs on a router is when you have to convert from a TDM trunk (FXO or T1/PRI) to an IP resource. Since SIP is all IP-based, you shouldn't have to do any conversion.
Do be aware that any conference calling features will consume DSPs unless you have a hardware conference bridge.
It depends on whether or not transcoding is required. If you are using a codec in-house that is incompatible with your SIP providers gateway then yes. Otherwise, no.
Same with Unity Express voicemail. If the voicemail is using the codec you run in-house, no transcoding required. However if you are compressing voicemails (recording them in g.729, for example), then you will have to invoke the DSP to transcode. Otherwise, users can't leave VM's.