in the body copy of a Standard page, is it possible to include a template. rather than just a link.
e.g. within home page, another header / footer page.
I think this can be done by the use of designs, but I may be wrong…
Not in the sense of including 'header' and 'footers' as seperate designs. The design area has a nest content ability where you are able to nest the contents of another asset (e.g. standard pages), so you could conceivably do what you are asking, but I wouldn't really recommend it.
is it possible to include "Search page" into a standard page
Yes, using the Nested Content bodycopy. You can nest the contents of any asset within the bodycopy of another, by using a different bodycopy type. If you open the Div Properties dialog box for a bodycopy, just change the type to Nested Content and select the asset you want to nest. 
However, nesting forms (like Search and Custom Form assets) is currently an issue when Cache Manager is enabled. I believe this has been resolved in v3.8.
Also, if you want to have a search box available on every page of your site, you would use the Searchbox design area within your design instead.