In the "lock screen" dialog, the fastest option is 5 minutes. Is there any way to lock it immediately, or at least after one minute?

Quick...
Power key between camera button and volume buttons -> "Secure device."
Be careful not to switch off accidentally.

press the power button twice quickly
or just use the lock slider next to the 3.5mm jack

Both ways, you get "Swipe to unlock" screen, and you don't need to enter a numeric code. It's like "Lock screen and keys" in the Power key menu.
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Device lock, which requires you to enter a numeric code, appears:
1. after five (or more, depending on settings) minutes of blank screen;
2. after you tap "Secure device" in the Power key menu;
3. on startup of the device.

Numeric code is also required when you change device lock settings, to make sure you remember it.

Actually, I suspect it's more to make sure that this isn't someone ELSE who got their hands on your phone trying to change your lock settings. This way only the person who knows the password can change the settings. Like when they ask you for the password in almost any change-password form.

(It might seems counter intuitive, since you're not always changing the lock code, until you remember that one of the lock settings is to lock the device after "never".)