When my children were young I did take the blades out of my razors and put them in a safe place, 0ther than that the razor has always been cleaned dried and blade put back and ready for the next mornings shave.
I will remove my razor blade and and pat dry on my towel and then place it on a magnetic fridge business card and proceed to rinse my razor and also dry with towel and leave razor lose to dry in the open air. I sometimes change razors in my rotation and like to use the same blade usually and by drying the blade you do prolong the life of the blade IMO regardless if it is stainless. The stainless they use in almost all razor blades is magnetic so the properties more than likely are Ferritic steels of high chromium and lower carbon which over time make a oxides to seal the blade from rusting. Water is the enemy of most steels regardless.
Yes, I leave it inside the razor head but I always clean the razor at the end of each shave so it waits for me cleaned for the next shave.
All stainless blades get rinsed clean with hot water and put back in the razor. Carbon blades same treatment but patted dry along with the razor before putting them back in.
I always rinse, dry and store my blades back inter wrapper prior to the next shave. This enables me to not only change blades and/or razors between shaves but also avoids some of the rust problems I've seen on stainless razors in the past.