1) Yes, your table looks correct, the switch will update the MAC-address table when it receives a new frame with the source address from the frame. That way it builds the MAC table of what endpoints are connected to what interface.
2) Any switch that I've seen checked doesn't show it's own interface MAC addresses, nor the in-band management interfaces MAC address in the MAC Address Table. The reason is probably to keep the tables more readable. In your example, if both switches are managed in-band, the switches should show each others MAC addresses in their tables.