When using an Android bound service type, can the Server (service) initiate a request to the Client (Activity)?
Yes, but depending on your architecture it may be limited. If you're in the same process, you can pass the service an interface via the Binder that it can call when it needs to send an event.
If you aren't in the same process, you can use AIDL. It's a pain, and its limited in what data types and data sizes you can pass.
You can also use BroadcastReceivers, but that's even more limited than AIDL, although easier to set up. This does have the advantage of being able to send to multiple clients easily.