1) Compatible SIP Server means a SIP Server that supports "late forking" i.e. a server that can fork an invite when a device registers, even though the device was not registered at the time the invite was sent. (This is required because if you are using the push notification strategy for mobile VoIP apps as required by iOS 10 and above, the mobile device will not be registered until after the device receives a push, and the push will be sent at the same time as the invite)
2) Ask the server admin, or if that is not an option try deregistering your device, placing a call, and re-registering your device and seeing if the invite comes in
3a) Kamillio has a Module called TSILO that supports late forking here is a video that both explains what the push notification strategy is and how to implement it in Kamillio using TSILO
3b) Linphone has an Open Source proxy server called Flexisip that supports late forking and can be used on top of your current server if it doesn't support late forking. Flexisip can be installed on your own server or on a cloud platform. Like Linphone, Flexisip has a dual license, free AGPL or a commercial license
4) No it is not restricted to only Linphone's own servers.
Keep in mind that the server still has to send a push notification at the time of the invite(If you are using Flexisip and configure it correctly Flexisip will take care of sending the push notification).