Well, there's a challenge I'll take!! My replies are my own view, as I have never worked in the airline world.
Why are our airports not set up to serve the travelling public?
Airports were always seen as a 'service' and developed ad hoc over many years with no central planning. Also, in their defence, LHR has never been given the money to have a 'big sweep'. That is, to start almost from scratch. They have always had to stay within the central area. When they did break out of it (T4) it was already too small (not enough money spent on it?) and the inter-connections with the central area were/are disastrous.
They could never get more land and they could not close down one terminal for two years to pull it down and start again. This led to a long period of Mend-And-Make-Do. Then the Conservative govt decided to privatise it as 'they' would put the money in - but then hobbled them with the price cap. So, instead of having a capitalist system, they wanted the best of both worlds - to not put in central money and yet control it. So, inevitably, we got the worst of both worlds.
Why are so many airport employees lacking in interpersonal skills?
I think this stems from the time when it was a civil service department. Although it was privatised 20 odd years ago, it is almost impossible to change the culture of an organisation. The British Airports Authority had been in place for too long and were/are too set in their ways.
From civil servants to private employees was overnight but they were the same people doing the same jobs. As new people came in they learnt the old way of doing it. Yes, I know that they had new training but when they got out into the offices and operational areas, they would have found themselves surrounded by the old crowd. Naturally, not all BAA staff would have been in that category but my 30 years of working in a range of commercial and governmental organisations tells me that changing the culture is an uphill job.
So, it is my long considered view that the situation is (pun intended) terminal!! They do not have the space to redevelop and the cost of doing almost anything is now frighteningly high because of the decades of inaction. BA finally got their way to have their own place and have convinced themselves that they can do their own thing and give their pax a good experience. Sorry to always be the Cassandra, but I doubt that it will work out that way.
BAA will be split up but it will not fix the problem. The problem will be solved by people going elsewhere and then the airlines finally banging the table but this might need a new generation of managers, rather than the old boys club that has kept CAA and BAA and the old carriers scratching each other's back for too long. I sit to be corrected.
Why are there inconsistent rules for different flights?
Can you define that more closely? Do you mean with regards to check-in, security, boarding, luggage?