At the bottom of the page it says:
Important: Prices may vary due to time of purchase and route flown. The earlier you buy, the more you’ll save!
Therefore, you can’t find the specific upper bound price unless you specify the route.
After a phone call with customer services they explained that they were having a technical fault, therefore the boarding passes couldn’t be accessed and the seats couldn’t be confirmed.
It turns out it was indeed an aircraft change and the new aircraft had more seats than the old one.
Barring places where you need to do customs declaration at an intermediate point of your journey (such as when your final leg is intra-US or intra-Japan) and barring connection times in excess of 24 hours, your luggage will be checked through to your final destination with almost absolute certainty.
In your case, having to recheck your luggage on a UK–MUC–...
Some airlines even enable you to "check-in" multiple times online or in their mobile app, and give you the option of changing your seat each time you do.
Subsequent "check-ins" after the first one are really just updates of your check-in. If you look at your boarding pass, there's usually a "sequence number" (often marked SEQ, but sometimes just a 3-digit ...
Have done this so many times, with many, many airlines. I often check in online in case I'm delayed by traffic etc, at least I've bought myself some time - but if I get there on time, I too like to get the actual boarding pass.
Also for 18 months I worked for an airline in Australia, had to fly regularly and we'd regularly reprint, change, update and ...
There are numerous hotels near the Heathrow airport. Before booking you must check out for check-in and check-out times. You can also store luggage at the hotel and again come back at the check-in time. In the meantime, you can pay a visit to the nearest malls for a refreshing time.