Must you work with all qubits in a circuit when applying a gate? When looking at this question:
How to apply a 1-qubit gate to a single qubit from an entangled pair?

And other questions, it makes it seem like when you have a quantum circuit involving $N$ qubits, that when you want to peform a gate operation on a subset of the qubits, that you must still make a matrix that applies to all qubits.

Is this always true, or is it only true in the case where qubits are entangled?

In other words... is it only entangled qubits that we have to process together, or must all qubits in a circuit be processed together?
You can apply a quantum gate on any subset of the qubits, not affecting the rest.

This can be formalized as evolving certain qubits via $U$ while the other qubits evolve via $I$, thus the joint unitary is $U \otimes I$.