I have some custom content type and custom page layout from that content type. This content type has HTML rich text field. And all my contents are page instances of that custom page layout. I need some "Create new" button/web part on the default page of the site, to give the content authors ability to create new content without going to standard sharepoint page creation process.
I don't know which part is your problem so here is some helpers:
To security trim a bit of HTML use SPSecurityTrimmedControl
To make your page layout default (or only allowed) click 'Page layouts and site templates' in the 'Look and Feel' section of site settings
SP.UI.ModalDialog.OpenPopUpPage(_spPageContextInfo.webServerRelativeUrl+'_layouts\u002fCreatePublishingPageDialog.aspx', null, 390)
To create a Publishing Page using code use PublishingWeb.AddPublishingPage