BuddyPress.org

In the BuddyPress settings, you could just turn off the Friend component.

1. Have you disabled site tracking? Try going to the WP dashboard where you can disable and enable BuddyPress components and check that site tracking is enabled.

2. Maybe check entries are recorded when using a different theme? Twenty Thirteen is a good tried and tested one.

In the BuddyPress admin area, you need to assign a WordPress page (created in the Dashboard under Pages/New) with a BP component. This codex page explains in more detail

Configure BuddyPress

The difference is in the scale and the level of complexity.

buddypress + WordPress on a single site is relatively simple to setup
wheras Buddypress+ WordPress Multisite is more complex and requires additional hosting resources

The functionality difference is on a single site install your users will be posting to a single blog whereas on a multisite instaall each user would have their own blog to post to.

If you are just seting up the site then my advice would be keep it simple and go for a single site WordPress+Buddypress setup

I’m a great believer in keeping things simple,and WordPress Multisite creates an additional layer of complexity which you probably do not require in initially.

The problem here is the bp_has_message_threads() function itself. It comes with hardcoded checks to override the $box to sentbox or inbox based on the results of bp_is_current_action().

You could try something this, in lieu of bp_has_message_threads():

function bp_custom_has_message_threads( $args = '' ) {
	global $messages_template;

	$r = wp_parse_args( $args, array(
		'user_id'      => bp_loggedin_user_id(),
		'box'          => 'inbox',
		'per_page'     => 10,
		'max'          => false,
		'type'         => 'all',
		'search_terms' => isset( $_REQUEST['s'] ) ? stripslashes( $_REQUEST['s'] ) : '',
		'page_arg'     => 'mpage', // See https://buddypress.trac.wordpress.org/ticket/3679
	) );

	if ( bp_is_current_action( 'notices' ) && !bp_current_user_can( 'bp_moderate' ) ) {
		bp_core_redirect( bp_displayed_user_domain() );
	}

	$messages_template = new BP_Messages_Box_Template( $r['user_id'], $r['box'], $r['per_page'], $r['max'], $r['type'], $r['search_terms'], $r['page_arg'] );

	return apply_filters( 'bp_has_message_threads', $messages_template->has_threads(), $messages_template );
}

this link might help you understand how the two work together:

WordPress Blogs

multisite is just activated by a setting in your wordpress config file it can be done before or after buddypress but if you are starting from scratch id probably do the multisite first, then add any plugins.

Copy it to your theme or child-theme

See ‘Overloading Template Compatibility theme files’ here:

Theme Compatibility & Template Files

The easiest path is just to add a conditional to an override of this template:
buddypress\bp-templates\bp-legacy\buddypress\members\single\profile\edit.php

The name field is, I believe, always:
input type="text" name="field_1" id="field_1"

So in the template, change this:
if ( 'textbox' == bp_get_the_profile_field_type() )

to:
if ( 'textbox' == bp_get_the_profile_field_type() && 'field_1' != bp_get_the_profile_field_input_name() )

and add an ‘else’ to just show the name value – bp_the_profile_field_edit_value()