ios push practicaly equals frequent requests to the servers? How iMessage push works? I would like make sure, how push in iOS works. Is it just about frequent requests to the servers?

Second question is how iMessage works? It is free of cost, works over the Internet (using the data transfer), but it is also based on frequent requests to some server or it is somehow classical GSM BTS backed?
About Push Notifications

As its name suggests, Apple Push Notification service (APNs) uses a push design to deliver notifications to devices and computers. A push design differs from its opposite, a pull design, in that the recipient of the notification passively listens for updates rather than actively polling for them. A push design makes possible a wide and timely dissemination of information with few of the scalability problems inherent with pull designs. APNs uses a persistent IP connection for implementing push notifications.

iMessage uses the same protocol as APNs (source), so it is pushed from the server, and does not use GSM.