re: Search form, to have a specific to search based on the field setup in settings.
There is no shortcode to display the search form. That said, what can be done since you have the Pro Pack. The Widget Pack once installed will add a search widget. If you add the Search widget to the Connections Widget Shortcodes sidebar. After configuring it and clicking the Save button you will be given a shortcode that you can use one a page.
re: How to make sure that the user profile on my site is one, means same for connection and same for wordpress site.
The Connections entry and the user profile data are completely separate and can not be combined on the same page. However, user the Link extension the two can be linked so the user can edit “their” own entry. Most users of Connections simple use the Connections admin profile page added by Link as the profile page. There is no shortcode for this as it exists in the admin just like the WP User Profile page.
Hope that helps!