Can fields validation errors be displayed below the fields?
I think you are looking for something like the Inline Form Errors Module. Here is a quote about it (from the module's project page):

... allows you to place form submission error inline with the form elements. Three options are provided for setting your inline error behaviour. You can configure the default behaviour or override the behaviour on a per form basis. You can add as many forms as you like.
IFE provides three behaviors for the configured forms

Leave the messages in place, this option will copy the error messages and place them inline. The original error messages set by Drupal will remain in place
Show an alternative message, this option will replace the original messages with a generic error message such as 'Please correct all errors.'. This message can be set in the IFE configuration page. The original error messages are placed inline with the form elements
Remove all messages, this option will remove all error messages and place them inline with the form element

In all cases only the messages related to the form will be touched. All other messages will remain in tact.