how to insert textbox value in existing database table in wordpress?

Though you haven’t mention the process and WP tables to insert textbox values in existing WP table. Either you can used option table or postmeta table for your purpose.

1. For option:
   WordPress provides two specification API functions for writing data to the database. One comes in the form of adding information, one comes in the form of updating information.

   • add_option
     1. A key – or a unique identifier for the information
     2. The value of the data to be stored

As per your sample form

if ( isset( $_POST['showCat1QuestionValue'] ) && ! empty( $_POST['showCat1QuestionValue'] ) {
    add_option( 'cate1', $_POST['showCat1QuestionValue'] );
}

• update_option

  1. Add the option if it doesn’t already exists
  2. Overwrite the existing value if it does exist

2.For Post meta:
Post meta API functions take in three piece of information:

1. The post ID
2. The data key
3. The data value

Sample code

global $post;

add_post_meta( $post->ID, 'cate1', 'showCat1QuestionValue');

Have a look for metabox in more details: