If you attach to a network successfully, it means the its secret key stored in the SIM card matches that of the network.

Since your IMSI catcher is basically just a proxy between the mobile phone and the real network, you can therefore easily detect that the IMSI is who it says it is.

No, because he who collects isn't interested in random mobile phones, and whoever tries to spoof "other" phones doesn't know the phones of interest of the collector. And the collector is interested not in knowing the presence but in the whole traffic, so the spoofing is even more obvious.