[SalesForce] How to use Chatter in an external Web Site

I want to have it where if someone makes multiple field updates in one edit, it posts to chatter like this:

This is actually a pretty simple change. See where you have

        FeedItem post = new FeedItem();
        post.ParentId = objNewAccount.Id; // RecordId
        post.Body = UserInfo.getName()+' changed '+fieldLabel+' from '+oldValue +' to '+newValue ;

        lstFieldChanges.add(post);

inside the inner for loop, which iterates field by field? You want that code to be in your outer for loop, that iterates Account by Account.

However, you're also wanting to accumulate a post body that mentions all of the changed fields. You can do that using very similar logic to what you're working with now. The way I like to do it is to accumulate a List<String>:

 List<String> changeDetails = new List<String>(); // at top of outer for loop

and then in my inner for loop

 changeDetails.add(fieldLabel+' from '+oldValue +' to '+newValue);

This allows you to efficiently template the final post body with String.format() when you're back at the bottom of your outer loop:

post.Body = String.format(
    '{} made the following changes:\n{}',
    new List<String> {
        UserInfo.getName(),
        String.join(changeDetails, '\n')
    }
); 

If also possible, can I add to the trigger so that it posts to a specific group or user? The reason why I'm writing a trigger is that we need to track more than 20 fields on a record.

Sure. Just change the ParentId of the feed item. You may want to amend the message to include details of the changed record. See the FeedItem reference under ParentId:

ID of the object type to which the FeedItem object is related. For example, set this field to a UserId to post to someone’s profile feed, or an AccountId to post to a specific account.

Yes, go to the Page Layout for the object, and then remove everything you don't want in the "Quick Actions in the Salesforce Classic Publisher" area.