If only half the password is written down, anyone who obtains that paper without knowing the algorithm only knows half the password!

Oh, that's right, thanks!

Or, you know, use a password manager.

Which password manager is the most secure / robust / potentially long lasting company out of: LastPass, 1Password, Dashlane, Keeper (or others?)?

I use LP, I also used 1Password professionally and I found it cumbersome but your mileage might vary. I disliked Dashlane and never used Keeper. They all do roughly the same thing, the difference is in UI mostly, just test it out and see what you like best.