More tabs for posts and comments

Do it just like AnsPress adds all other tabs. Look in sources: ap_register_user_page( 'about', __( 'About', 'ap' ), array( $this, 'about_page' ) ); ap_register_user_page( 'activity-feed', __( 'Activity Feed', 'ap' ), array( $this, 'feed_page' ), true ); ap_register_user_page( 'notification', __( 'Notification', 'ap' ), array( $this, 'notification_page' ), true, false ); ap_register_user_page( 'profile', __( 'Profile', 'ap' ), array( $this, 'profile_page' ), true, false ); ap_register_user_page( 'questions', __( 'Questions', 'ap' ), array( $this, 'questions_page' ) ); ap_register_user_page( 'answers', __( 'Answers', 'ap' ), array( $this, 'answers_page' ) ); ap_register_user_page( 'followers', __( 'Followers', 'ap' ), array( $this, 'followers_page' ) ); ap_register_user_page( 'following', __( 'Following', 'ap' ), array( $this, 'following_page' ) );

Who creates tags on the questions submitted – the user that submitted the question or the administrator of the site?

If there is no category, then category fields will be hidden. But user can add new tag if that doesn’t appear in tags suggestion.

Can you set up AnsPress so every question submitted by a user must be approved by an administrator?

Yes, there are option for you need. Check in wp-admin->anspress->options

How can I add new action/feature to questions and answers?

I agree with @Dima. There is no need of additional like system. You can easily turn voting button to like button. There is an option in AnsPress to hide down vote, do it and add some CSS.

2.4.1 – Two un-needed links to each Question

.ap-q-inner .ap-post-history { display: none; }

How can I add new action/feature to questions and answers?

You would need to write custom PHP code to achieve this. It will be pretty simple, just mimic the upvotes logic. (are you sure you want to add another likes system? there is already “likes” and “dislikes”, the thumbsup and thumbsdown icons near the number)

How to get vote counts for a particular question and answer?

var_dump(ap_post_votes(ap_question_get_the_ID()));

Cannot submit a question, recaptcha error

I have upgrade my php to PHP Version 5.6.15 and now it`s ok

Can you set up AnsPress so every question submitted by a user must be approved by an administrator?

Just spend a few minutes to review the options. In Anspress admin / Options / Moderate. It is SO evident !!

Cannot submit a question, recaptcha error

This issue has been reported already and I am trying to find the issue.Is there any error in your log?