TDM=Time Division Multiplexing, which is how GSM works. Different operators have to build their own network, and when you're with one operator, you can't use another operators network. The exception to this is emergency numbers, which will work on a GSM phone even without a SIM card. In general, you receive a new SIM card when you switch operator. The SIM card holds the information needed to authenticate with an operators network. The frequency, modulation, and the whole shebang remains the same between operators, in fact, with most phones you can search them up, but probably not connect.
GSM does not have line cards, in the traditional POTS sense. Phones have a unique key, and so does the SIM card. This is how the operator keeps track of its subscribers.
The connection between one operator to another is done in a central switching system, yes. Operators switching systems a connected together.
But then there is roaming, but that is a whole other story.