IOS – How to make the App Store remember the password

There is no way to stop the 'last key ghosting' that you mention, it's baked into the OS and not controlled by a setting anywhere. A jailbroken phone may have possibilities if you are happy with hacks etc, but I don't think you are interested in that route (correct me if wrong).

What you can do is tweak the passcode requirements in ways that may reduce the number of times you need to enter your passcode. By default, when you lock your screen (or when it times out and does it itself) you need to re-enter your pin when unlocking. There is an option in Settings > General > Passcode Lock called Require Passcode with options of Immediately, After 1/5/15 minutes.

This is very useful if you want to quickly unlock after it timed out on you without needing to retype your passcode, and does not require you to turn auto-lock off.

Realise this is only a workaround for half of your problem, but may prove useful regardless.

As for other passwords, you could keep them in a password manager program which allows you to copy the passwords whilst keeping them obscured and paste them into password fields. I don't have any specific recommendations, but leave it to other answers who may be able to take inspiration from experience with these apps.

Enabling automatic downloads can help, but it's highly unlikely that this is simply a preference on the client. I would presume (but haven't reverse engineered) that the server is signaling the purchases that require a password either by signing a certificate with a short expiration time or upon each "purchase."