In response to your 2nd question, the answer is - it depends. If there is VLAN tagging between your two switches, then the L2 header would be modified with one or more VLAN tags on the connection between the two switches (however, this is not to say that any MAC addresses directly associated with the 2960s themselves would be involved in any kind of forwarding decision between the PCs - they wouldn't be).
Assuming VLAN tagging is not being used in your example, then the answer would be no. Each switch would only know that there are multiple MAC addresses learned on each of its interfaces that have the connection to the other switch.
The top 2960 would have a MAC address table with 4 entries, and those entires would be the PC MAC addresses. One PC MAC on each of those ports, and 2 PC MACs learned via the interface that the bottom 2960 connects to.
Similarly, the bottom 2960 would also have a MAC address table with 4 entries. One per PC MAC that was directly connected to the switch, and another 2 PC MACs learned via the interconnected interface to the top switch.
Edit: And for completeness, as Ron pointed out, the L2 header is not modified when the frames are not crossing an L3 boundary.