There is no such thing as 'correct' care IMHO - simply what you want to do (within reason)
Yes
No need, unless you want a sharp object lying around to cut yourself with.
Why? The blade will cope with the tension - these are MAN blades after all - not those sookie cartridge things!
In all seriousness there is wisdom in the comments from
@alfredus,
@nsavage and
@thisISjoel BUT ...
I've also got and have cleaned up many razors which are up to 113 years old. I've seen them in all sorts of condition and they almost invariably clean up pretty well. The only things that really cause damage are drops and plate loss. Personally, I think I would be more likely to damage a razor by dropping it when I'm trying to clean it than by just rinsing off excess lather and leaving it to dry with blade in place on the vanity - but that might just be me being a klutz.
If you are more dexterous than I am - go for it, clear to your hearts content after each shave to whatever standard you want. I'd suggest though that the result might be that it looks like after 250 years rather than starting to deteriorate after 200 years!
None of the above applies to Zamak razors which I classify as expensive disposables. Show me one of those after 100 years
(cantankerous old bloke today aren't I)