Hi,
for clarification, you can change into Neutral whilst moving (although not sure why you would want to) you must NEVER go into Park whilst moving. As James said, park engages a physical lock which is why you can use it instead of a handbrake (as most auto drivers do

) so if you go into park whilst moving the box will go BANG.
To be fair 82JEW75 said go from drive to reverse not park, but i still wouldn't do this on this or most auto boxes ever, unless you know your box specifically allows it. Some of the later electronic boxes, for example do, it used to be a cool trick when selling Chrysler Voyagers that you could be driving along and flick the transmission into reverse at any speed

- I never had the nerve mind you, but was shown it by one of the factory engineers in Austria - the box was effectively "fly by wire" and the electronics stopped the box from actually changing.
Back to your problems, I don't want to rain on your parade (which I feel like I am doing) but be VERY careful if you plan on doing a "strip, clean and rebuild". Most mechanics won't touch auto box's as they can be a bit complex.
You can drop the sump, change the filter and replace the fluid no probs. When you drop the sump you will see the valve body, if your really brave you can take this out and clean it but be VERY CAREFUL as it has little springs and ball bearings which will make an determined bid for freedom and once you loose them or don't put them back in the same place your box will never work again
You can check the pressure in the box and adjust the brake bands as described in Bentley but again that tends to be pretty specialist and not something your average mechanic does.
A full strip is a specialist job in an auto tranny repair shop, there are a few around but most of them have long since stopped doing these sort of boxes and got rid of all the special tooling. I keep an eye out for this tooling myself but have not yet come across any
Sorry if that sounds not very encouraging, just want you to make sure you know what you are getting into
Cheers,
MM