What does this flow mean? (SDN)

I think you've mostly got it.

There are a few points I would make though.

3. The path P_0, from R to C_1, is often (but not always) implemented 'out-of-band', as a separate network.

4. Only one, or a handful, of packets for each network flow should reach C_1. As soon as the flows are inserted by the AmyOSPF controller, they are simply forwarded according to that routing logic. This is what the second quote says.

5. No. AmyOSPF computes the best path P_1 from R to the destination of the packets, wherever that may be. It can do this without talking to the switches.

The flow entries that make up the path P_1 is then sent to the routers on P_1, so that all traffic coming from Amy's machine to a particular destination are sent along that path, without having to go to the controller. In the case of R, this new flow entry must have higher priority than the one that says to send all traffic from that port to the controller.

6. It's not really a multicast, the controller has a separate connection to each switch/router. The flow entries may be slightly different for each router too.

7. The point is that it doesn't matter what 'view' the router has, as all of the routing is done in the controller. But C_0 does have a global view.

This will have to be repeated for each new network flow originating from Amy's machine. It wouldn't make sense to create a path from R to X and then send packets destined for Y along it. Having said that, flow entries can be aggregated.

When the network is set up, the controller on recognizing the switch has been added to the network will install the default rule to ensure it gets messages from that switch that are not resolved by higher priority rules. All subsequent rules installed are considered higher priority.

but in real networks we cannot connect each switch directly to the controller.

In SDN networks all OpenFlow switches are directly connected to controllers. Recall that the routing intelligence is removed from OpenFlow switches and contained in the controller. An OpenFlow switch therefore cannot change its flow rules without that connection.

Traditional switches and routers do not connect to SDN controllers because they contain the routing algorithms that are centralized in SDN.

There are multiple ways to create the controller connection, but the principle remains that every OpenFlow switch talks directly to the controller which sits off the forwarding path.