Regarding to your first question:
As you said, the SWAP gate can be decomposed into 3 CNOTs as:
and the Bridge gate can be decomposed into 4 CNOTs as:
From the circuits above, we can see that the SWAP gate only involve 2 qubits. It Swap them. Whereas the Bridge gate is to perform a CNOT gate between two non-adjacent qubits with one qubit between them.
But also note that:
This is important because suppose your device have the following architecture:
which doesn't have qubit-0, $q_0$, connected to qubit 2, $q_2$, but you need to apply a CNOT gate between them in the middle of the circuit you are executing. Then, instead of naively using the SWAP gate to swap qubit 1,$q_1$, with qubit 2, $q_2$, then do a CNOT on $q_0$ and $q_1$, then apply another SWAP gate to swap qubit 1 and qubit 2 back to their original spot, we can just use the Bridge gate.
And you can combine both gate together too. For instance, you want to execute a CNOT between qubit 0, and qubit 4. For the above hardware architecture, you can SWAP qubit 3 and qubit 4, perform Bridge gate, then SWAP qubit 3 and qubit 4 back again. That is,