Do we use link layer switches only in LANs, or are they deployed in the network core as well? I cannot picture the network core in my mind, is it just all routers starting from the edge router?
Your question is really too broad, and may need to be closed because it may start generating primarily opinion-based answers. There are also newer switches and topologies for data centers that do things much differently.
Cisco has (had?) a three-tier model model where the layer-2 access switches connected to the layer-3 distribution. The layer-3 distribution would then connect via a layer-2 core. The idea is that the core tier simply switches as fast as possible, rather than spend time routing, which is what is supposed to happen in the distribution tier.
That can be applied differently, depending on the size and needs of a site. For example, a medium-sized site could collapse the core and distribution into some chassis switches, while a small-sized site could have all three in a single device that combines a router with a switch module.
Another change is that the current best practice is to run routing on the access switches. They would connect via layer-3 to the distribution, rather than the traditional layer-2 connection to the distribution. They would still have layer-2 connections for the access interfaces. This practice eliminates many layer-2 problems that could spread (STP problems and broadcast storms), isolating any layer-2 problem to the single access switch.