I don't see a need for special care unless one is using a carbon steel blade.
None, I leave it in the razor till it starts to tug and then replace it.
^^^^^^^^^^^^^Rinse the razor well, flick loose water off the razor, and hang it on the razor stand.
Let's just note that most shaving soap has conditioning ingredients in it as well as plain soap: extra fat, shea butter, almond oil, cooonut oil, lanolin, glycerin, etc. Plus chelating ingredients that cause minerals to precipitate out of the water. These things leave deposits.Soap is soap.
It ALWAYS cleans.
It's highly unlikely for your blade to rust, most of them are stainless. If it was going to rust it would do so the first shave it wasn't thoroughly dried...With my shavette, I just rinse off the blade and give it a quick wipe with a towel. I only use the blades twice before I pitch them, so I dont worry about them rusting.
When I used a DE, I used to always rinse the razor and then give it a quick dunk in a jar of alcohol to displace any water and prevent rust. Some people like to disassemble their razor and give it a full wipedown but Im just not going to do that when I can dunk it in alcohol and accomplish the same thing.
They can still rust. Theyre stain-less, not stainproof. Back when I used a DE, I had plenty of stainless blades that would rust after about a week before I started doing the alcohol dunk method. The same when I used carts and disposables: after about 2 weeks that blades would develop a bit of rust.It's highly unlikely for your blade to rust, most of them are stainless. If it was going to rust it would do so the first shave it wasn't thoroughly dried...