TL;DR - the ticket numbers are an artifact of historical processes, and you should not encounter any issues with connections. Checking bags all the way through will be subject to airline interline agreements. Get the second airline's PNR from your travel agent to change seats on those flights.
For context, a Passenger Name Record (PNR) is the reservation of inventory on flight segments, and a Ticket is confirmation of payment for flights. Historically, the check-in process was a means of verifying that the reserved inventory with tickets to reconcile that the passenger had, in fact, paid for the flight. The IATA ONE Order initiative seeks to combine these two independent records into one and solve many traveler headaches in the process (if your ticket and PNR ever get out of sync, it is painful to fix, based on personal experience).
At the time of this response in 2020, tickets can only contain up to four flight segments.
I once heard a story that suggested the original ticketing systems constituted paper forms that were printed with space for four flight segments, since at the time (before computers were used to track flight inventories and sales), the vast majority of tickets did not need more than four flight segments. According to this blog, as of 2017, the average number of segments per ticket is only 2.5.
To overcome the four segments per ticket limit, travel agents/airlines issue conjunction tickets, sequential ticket numbers that cover each group of four flights, in order of time. (i.e., the first four flights will fall on the first ticket number, the next four on the second ticket number, etc.) Section 184.108.40.206 of the IATA Ticketing Handbook has some details about conjunction tickets.
As of 2020, most systems are limited to issuing four tickets in conjunction, leading to the common 16 flight segment limit on round-the-world tickets.
Because your flights are booked under a single PNR, the contract of carriage will continue throughout your entire itinerary, irrespective of the ticket number. If you were to misconnect at any connection due to a delay, the airlines involved are still required to convey you to your final destination for those flight segment(s).
In order to handle passengers connecting between airlines, airlines often engage in an interline agreement. If the airlines you are connecting between have an interline agreement, then you will be able to check your bags through. You mentioned you are flying KLM and China Southern, who appear to have an interline, based on the KLM website.
Regarding your question about selecting seats on China Southern, if the airlines in question run different reservation systems (the largest are Amadeus and Sabre), then an additional PNR will be generated for the second airline. If you call your travel agent or issuing airline, you can ask them for the PNR for the other airline, which is often required to change seat assignments, unless the second airline has implemented a method to find their local PNR based on another system's PNR (I generally only see that within alliances).