set Timeout element

Using AutoAction, I need to disable element for 24h. I have tried various triggers like onBeforeUserProfileEditDisplay and even onAfterUserProfileSaved but no matter what I choose when the user edits profile again, he can change the value of the element before time set expires. Is there a way to keep the setting to disable until 24h have passed?

That's just a JS timeout. That's not going to have any impact once they navigate away from the page and it won't prevent saves server side and should be considered entirely cosmetic.

If you want a 24 hour lock on a field you'll need to use something like CB Conditional (be sure to set Reset to Disabled) and have an actual condition to block editing of the field. You can use a Code condition type to write custom PHP to check against last visit date or last update date and see if its been 24 hours since they last logged in or last updated, but ideally for this to work best you'll want to keep track of when that individual field was updated.

Is the last modified date of a field stored somewhere in the database?

No, we don't keep track of individual fields modify dates. You'd need to implement that yourself using a custom database table and CB Auto Actions. Use the below to have an auto action act on field value change and is when you'd insert/update that database table. You can then use a Query condition to query for its update date.

Global
Triggers: onAfterUserUpdate
User: Automatic
Access: Everybody
Conditions
Field: Custom > Value
Custom Value: [var1_FIELD_NAME]
Operator: Not Equal To
Value: [var3_FIELD_NAME]

Replace FIELD_NAME with the name of the field you want to check has changed. The triggers used only applies to frontend profile edit. If you want to cover frontend/backend registration and frontend/backend profile updates you can use the below triggers.

onAfterUserRegistration, onAfterNewUser, onAfterUserUpdate, onAfterUpdateUser