I'm puzzled as to how search_path should be used,. Should all references be schema qualified inside functions body ?
Pretty much...you can also do:
CREATE FUNCTION funcname()
SET search_path TO 'other_schemas_needed_by_this_function'
AS $$
[...]
$$
You don't have to specify the schema the function is going to reside in...but there is exposure if you don't.
Or is search_path safe except in the body of functions used in index or constraints ?
pg_dump/pg_restore tends to be very conservative in setting search_path. I'd say you are safe if you can successfully dump/restore and unsafe if you cannot.
Cross-schema dependencies can be problematic and if you are not willing to test that your omissions are immaterial I'd say you should take the paranoid route an schema-prefix everything - either explicitly or by taking advantage of attribute setting options for functions.
Views, materialized and otherwise, are other areas commonly affected by lax schema specifications.