The conducting electrons are for the most part localized in the copper oxide layers. That is why condensed matter theorists (and experimentalists) who study high Tc and related phases (pseudo gap, strange metal) or spin liquids (which people think are somehow related) are so interested in 2D systems. In general 2D systems have very interesting transport properties (graphene, transition metal, dichalcogenides).
The behavior of the cuprates in a magnetic field is more complicated since the sc coherence length is actually smaller than the London penetration depth (where you get the Meissner effect). You will start to get vortices (Abrikosov) at a certain field strength. Basically the vortex is a super current swirling into a nonsuperconducting region and is like a magnetic monopole since it gives one flux quantum from the field of the super current. The vortices can form a lattice. Eventually they will condense and destroy the sc state at the critical field strength.