Use of Gravity Form as email optin form in sidebar & below the post

The first thing i'd say is I wouldn't recommend displaying the SAME form multiple times on a page. It can cause some strange behavior depending on features and functionality you may be using. Especially when implementing more complex features that utilize JS. That doesn't mean you can't do it, it just means that it's possible you may encounter strange behavior.

There is no step by step tutorial that explains how to do everything you have asked. You've asked a few different questions so each one of them have their own documentation associated with how to do what you describe.

If you would like to display a form in a widget then you'd simply add the Gravity Forms widget to the sidebar and configure it's settings just like any other widget.

If you want to display a form below a single post then you would simply insert the Gravity Forms shortcode to display the form you'd like to display at the bottom of the post content when creating the post.

Here is documentation on how to embed a form in a page or post using the shortcode:

http://www.gravityhelp.com/documentation/page/Embedding_A_Form#Page.2FPost_Editor

If you want to display a form below EVERY SINGLE post globally and therefore don't want to actually put the form shortcode within the page or post content itself then you would need to edit your theme and implement displaying the form using a PHP function call.

Here is documentation on how to display a form using a PHP function call within a theme:

http://www.gravityhelp.com/documentation/page/Embedding_A_Form#Function_Call

Now please note. When displaying a form via a PHP function there is more involved. You need to also enqueue the necessary styles and scripts. This is handled automatically when displaying a form via a shortcode, but not when doing so with a function call because of how WordPress enqueue functionality works.

Here is documentation on how to make sure you enqueue the necessary CSS and scripts for a form when displaying it via a PHP function call within your theme:

http://www.gravityhelp.com/documentation/page/Gravity_form_enqueue_scripts

As for MailChimp, you would integrate MailChimp using the MailChimp Add-On. The documentation for the MailChimp Add-On explains how to implement MailChimp on a form. You can find it's documentation here:

http://www.gravityhelp.com/documentation/page/MailChimp_Add-On

The documentation covers the questions you asked and the main documentation can be found here:

http://www.gravityhelp.com/documentation/page/Gravity_Forms_Documentation